From b22b5a999f8a858dd3d4cdfb125d4e880c966ef3 Mon Sep 17 00:00:00 2001 From: Peter Schaefer Date: Wed, 3 Apr 2013 12:27:47 +0200 Subject: [PATCH] [doc] kleine Fehler korrigiert [doc] plots aktualisiert [src] Code ausgemistet --- doc/code.pdf | Bin 234752 -> 366358 bytes doc/code.tex | 68 +- doc/doc.pdf | Bin 447281 -> 448705 bytes doc/doc.tex | 27 +- doc/fig/132t05n05_2DQuad_cond.eps | 878 +++++++++++++++ doc/fig/132t05n05_2DQuad_error.eps | 1324 ++++++++++++++++++++++ doc/fig/132t05n05_2DQuad_hminmax.eps | 1395 ++++++++++++++++++++++++ doc/fig/132t05n05_2DQuad_time.eps | 852 +++++++++++++++ doc/fig/1t05n05_2DQuad_cond.eps | 38 +- doc/fig/1t05n05_2DQuad_error.eps | 2 +- doc/fig/1t05n05_2DQuad_hminmax.eps | 2 +- doc/fig/1t05n05_2DQuad_time.eps | 2 +- doc/fig/1t05n05_3DFichCube_cond.eps | 38 +- doc/fig/1t05n05_3DFichCube_error.eps | 2 +- doc/fig/1t05n05_3DFichCube_hminmax.eps | 2 +- doc/fig/1t05n05_3DFichCube_time.eps | 2 +- doc/fig/1t05n0_2DQuad_cond.eps | 32 +- doc/fig/1t05n0_2DQuad_error.eps | 2 +- doc/fig/1t05n0_2DQuad_hminmax.eps | 2 +- doc/fig/1t05n0_2DQuad_time.eps | 2 +- doc/fig/1t1n0_2DQuad_cond.eps | 32 +- doc/fig/1t1n0_2DQuad_error.eps | 2 +- doc/fig/1t1n0_2DQuad_hminmax.eps | 2 +- doc/fig/1t1n0_2DQuad_time.eps | 2 +- doc/fig/1tn_2DQuad_cond.eps | 78 +- doc/fig/1tn_2DQuad_error.eps | 2 +- doc/fig/1tn_2DQuad_hminmax.eps | 2 +- doc/fig/1tn_2DQuad_time.eps | 2 +- doc/fig/exmpl11_ref.eps | 4 +- doc/fig/exmpl12_ref.eps | 4 +- doc/fig/exmpl13_ref.eps | 4 +- doc/fig/exmpl_2DLShape_ref.eps | 4 +- doc/fig/exmpl_2DQuad_ref.eps | 4 +- doc/fig/exmpl_3DCube_ref.eps | 4 +- doc/fig/exmpl_3DFichCube_ref.eps | 4 +- doc/fig/net1.eps | 4 +- doc/fig/net_single.eps | 37 +- doc/fig/net_wrong.eps | 4 +- src/A_plots.m | 2 +- src/areaQuad.m | 2 +- src/compute.m | 91 +- src/export_exmpl.m | 17 +- src/refineQuad.m | 61 +- src/test_sol.m | 2 +- 44 files changed, 4697 insertions(+), 343 deletions(-) create mode 100644 doc/fig/132t05n05_2DQuad_cond.eps create mode 100644 doc/fig/132t05n05_2DQuad_error.eps create mode 100644 doc/fig/132t05n05_2DQuad_hminmax.eps create mode 100644 doc/fig/132t05n05_2DQuad_time.eps diff --git a/doc/code.pdf b/doc/code.pdf index 1174f1ba63882838ac866a3b461797fb8305b0b5..fbc4e11c2ec3f02a70c4a3a4c051ea1ee4c70b55 100644 GIT binary patch literal 366358 zcma&NQ;;rAx3%52ZLYR$t+wse-K)*JZQHhO+qP}nw)=nHH@>}3o`Zj%RAgk;K~+SJ z%xjD}lT1NGjE;$(6^3l&GA18}iHnGl$i~0|hL@K?#@O1_(Ts?h@t-Lr1~GFhM`L>; z1~DsrM`ICVLmMMwetsASM|)#^0E}zqRjRZNDihKOSIsLDzq`+lUJT@qBjlCr36 zDQ9AHV=1fGFVRr>naGBiLi2DlXe6reM!U;A) zZq(Z3At9Vmci|G&3z`MzONoa@p7UKzmwSJWgu|*&U1>{G4Y+Vi{mA<@8k!0Td46v6 z0w`_~6vmHtwVu-_m!ZSs=fYDjB!$QATSH5mjPCcEp$3Bz28apJ-b{%MARu6k71l^* zC#2Uwep_k8BDT}Z#LLV3)bivR4!8N6^OK15{GCtFos`~b2_ZlA_fWOx@Eu>mGb;Y3 zW*D8I$%<3ezitc>77I3)j{L`~Sr8 zwuQ9J8?&ED_z%p)lWLGs)NCA+$u{8#2`s702Pc zhM5M&bSzm2rr3~x4Mv?TZ5Bxg$yeGAXAhMuNX`W9Ij(hMi5VVO(%=y%P=3Yf!<);! z%nl9_om^7btaA=5b%HD})U1nh<2g6tJ_8dvqwl%jI=$wdJDjJ~W9c8G={qErH+VnV zohO;BD@4MLvh+=B@;?|_E3us(HX51=YAT*ySW0H>ThI5lG4d=ajjD31Oe16_QK_XM z&M0LJ(9l1^D?w0l;0$Koo_gyrrCk=-Xad0J>vAT>)<*wNhX0KIquu{1_CEtoE*6;o zH^<8IFJZH?{-1>X=geVa82!fd8$I+3TxM!1n=`#onMw_(9->s;T1xxFYo-w40i zNdD)Ae)Ei*9Qa~= z>rHh3v*$)(W10DwkUcANCg^^U;5YgaG7l>|xIc=k2dDdCz=`rZm{#HY*tRe&G!psS zku&~Tvdr*@mQ*zN>npFQRSDw9n;Wl;=r4EVCt=8DG%9u8m-Gg;2HM4}fPeRhvo9l( z-J|oT^mOjuIwq2Gc-Y#R;kG@f4Y+j# zI)2AdB*JK-x#i9BzF8L4KN-~C?H9Rtc)9SdzJ^dh;0jRuSbaSgd_C;9y@d2V!`hVS zsvr~ul?G;!&pP7_pl*V(E$l3II=rW(K<%fJs9=)Xn%K8nzyP&6-K+Yn(b7N8oLU5t%Non3rS#Car#29 zV-fj9!^!%4UiTNfsD0~!4*F|~`wqp?hR24X+oM?woGGYY#3eH?30Z0>1!VG)s4t)R zM&y>yQ28v=i2iYvOM#s|H6*$bqVtBl2O8xen$b5VpLG|BzReS1o}i@r5`KLPCqZ3S zMPo)^?9^#)CD-vD*ON)$7S{2a_a%#tbn6NOk6@+%>wNZ&*3 zp0lq?WJUf^*$nXP=c$I^>+J)?4C`xTWk(|l%*%S7kr!}B37rbC6$PDBOh}5XIFpf0 zEHYgLOKX-6K_gL<5&h1A;rywpHWeZv%VRUtN>a{K2+Wrgn$nHm7yaEu?&on+*##3h zWy&2#z&<17a*SOFsWa;@n`xXGx9zMm`R4bc_fJrmq*ore$6Pc7qv_iHWSc!2f%* z>pp#m?}Yd9L#{IwmqI}P^xaVvVcu0phIubpKTzqKNxKO<2hrnkbaE|!_ z)qyCOANrQs`;7q&08?kBz9+*}t&epC*sArj#l*8qIyk|)R)IS*^{eSX=A#b@oBpyo z_A4`7)u2dLdyE&~zy2Nj#x;y`%~7jVO4Lou_SShv)(_WxrSAi0`+*tT1005TTftn1 zCv?DD&+^7mZQyWL{iJ#r=qLCM*Vbow%0&Uc5y-BBp9??P{lJ?SQ!P7cHRUN>r*py$ z8$Z1|+u3s6`PNE!f=!gpl1M!-cN<3@DKpkSOvnDmcjFjkiKMlPMPF;(?=ny>MKr$~ zE{agHw!KUl*lH12)Lw${^zgznTZX2eDfq7RXtH#!ly#+64OuScyQnR5sofYY(0G$H zj?r#(KTj4#@Z-=B@mC;nW$fML1*9ai-En`pb`}GRb*zNLkr8?rQI;P4!Uu?BcZ6Vm z<(t24IxMMP=rqR)ikdsL>&p_I6*_lgFuocAof*b>A|sd~OPC9&2I|nK{Ec)7(%m zX(!n$jX|Riwo9>s%|HHTTa-6Itcvx%XuxE!D@GyM8m(typ$SlhekXy|Byq%>TPYtX zIg$#qOD!DUZ>v~Z;O9Ua=&mD$*1HCbou&a0$&bB1;Jx06LGj?u#^GnW!laP*RiI4f zC}Zs06!tl(-b=Xg1#J^=R$<1HoKnvlsIM=n85}9*YT<&~TQKDNAZrF`CR8cq-IOWs zaU!huI2s-_tF-O2PD7-#+f7e}jUSouuzulfj%oYB=~WF;Lj7dp1Qr&0iPDZnQmrD6 zy1;JmM)`e~xn18OWZCB<8DoR$PsdZq+D-AEx!Ymh<@>LJDzu4PAAD;DfEt?~@y@|> zwpv|15Oq!C2)5c9NRP8_OA&9r)ebe#Mwi-_IesT~8M3Y|leMBq47#xXV-}(;7_+(r zEio*!i6r`oc1W{uC~z|1ki$wA?Eoq_C4i{Dy%O3CkR@mYO6_RCLL9XhOmKBnp6e%! z1e(zo_^5I{X!;b_W7%)m2#JqKqnCIb8uaQ~VzZjL-t(ZNBHwn%Q<0E4#m&fg?!Um=E`ZrLg9(k0?TF!jO&6-g(M?VJVMsoO5fOg}ipmFRD5zcng=wPKJ? 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+\psfrag{N-14}{\tiny $N^{-1/4}$} +\psfrag{N-34}{\tiny $N^{-3/4}$} + \psfrag{tmu 1t05n05 A}{\tiny $\tilde \mu$} \psfrag{eta 1t05n05 A}{\tiny $\eta$} \psfrag{fehler 1t05n05 A}{\tiny Fehler} +\psfrag{Zeit 1t05n05 A}{\tiny Zeit} -\psfrag{erste Fehler}{\tiny erste Fehler} +\psfrag{tmu 132t05n05 A}{\tiny $\tilde \mu$ analytisch} +\psfrag{eta 132t05n05 A}{\tiny $\eta$ analytisch} +\psfrag{fehler 132t05n05 A}{\tiny Fehler analytisch} +\psfrag{Zeit 132t05n05 A}{\tiny Zeit analytisch} -\psfrag{Zeit 1t05n05 A}{\tiny Zeit} +\psfrag{tmu 132t05n05 QEQA}{\tiny $\tilde \mu$ semiquadratur QE} +\psfrag{eta 132t05n05 QEQA}{\tiny $\eta$ semiquadratur QE} +\psfrag{fehler 132t05n05 QEQA}{\tiny Fehler semiquadratur QE} +\psfrag{Zeit 132t05n05 QEQA}{\tiny Zeit semiquadratur QE} + +\psfrag{tmu 132t05n05 QA}{\tiny $\tilde \mu$ semiquadratur Q} +\psfrag{eta 132t05n05 QA}{\tiny $\eta$ semiquadratur Q} +\psfrag{fehler 132t05n05 QA}{\tiny Fehler semiquadratur Q} +\psfrag{Zeit 132t05n05 QA}{\tiny Zeit semiquadratur Q} -\psfrag{N-12}{\tiny $N^{-1/2}$} -\psfrag{N-14}{\tiny $N^{-1/4}$} -\psfrag{N-34}{\tiny $N^{-3/4}$} \usepackage{lstings} @@ -62,15 +76,15 @@ \lstinputlisting[language=C++]{../src/slpRectangle.cpp} \subsubsection{gauss.hpp} \lstinputlisting[language=C++]{../src/gauss.hpp} -% \subsection{Matlab} -% \subsubsection{compute.m} -% \lstinputlisting[language=M]{../src/compute.m} -% \subsubsection{refineQuad.m} -% \lstinputlisting[language=M]{../src/refineQuad.m} -% \subsubsection{areaQuad.m} -% \lstinputlisting[language=M]{../src/areaQuad.m} -% \subsubsection{mark.m} -% \lstinputlisting[language=M]{../src/mark.m} + \subsection{Matlab} + \subsubsection{compute.m} + \lstinputlisting[language=M]{../src/compute.m} + \subsubsection{refineQuad.m} + \lstinputlisting[language=M]{../src/refineQuad.m} + \subsubsection{areaQuad.m} + \lstinputlisting[language=M]{../src/areaQuad.m} + \subsubsection{mark.m} + \lstinputlisting[language=M]{../src/mark.m} % \subsubsection{plotShape.m} % \lstinputlisting[language=M]{../src/plotShape.m} @@ -90,9 +104,10 @@ \end{figure} \begin{figure}[ht] + \caption{2D Quad adaptiv anisotrop vollanalytisch $V\phi = 1$} \centering -\label{fig:exmplAA_2DQuad} +\label{fig:exmplAA_2DQuad_A} \subfloat[Fehler]{\includegraphics[width=0.7\textwidth]{fig/1t05n05_2DQuad_error}}\\ \subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_2DQuad_hminmax}} \subfloat[Kondition]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_2DQuad_cond}} @@ -111,14 +126,14 @@ \psfrag{fehler 1t05n0 A}{\tiny Fehler (isotrop)} \psfrag{Zeit 1t05n0 A}{\tiny Zeit (isotrop)} -\psfrag*{tmu 1t05n05 A}{\tiny \quad (anisotrop)} -\psfrag*{eta 1t05n05 A}{\tiny \quad (anisotrop)} -\psfrag*{fehler 1t05n05 A}{\tiny \qquad\quad (anisotrop)} -\psfrag*{Zeit 1t05n05 A}{\tiny \qquad (anisotrop)} +\psfrag{tmu 1t05n05 A}{\tiny $\tilde \mu$ (anisotrop)} +\psfrag{eta 1t05n05 A}{\tiny $\eta$ (anisotrop)} +\psfrag{fehler 1t05n05 A}{\tiny Fehler (anisotrop)} +\psfrag{Zeit 1t05n05 A}{\tiny Zeit (anisotrop)} \caption{2D Quad im Vergleich vollanalytisch $V\phi = 1$} \centering -\label{fig:exmpl_2DQuad_tn} +\label{fig:exmpl_2DQuad_tn_A} \subfloat[Fehler]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_error}} \subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_hminmax}}\\ \subfloat[Kondition]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_cond}} @@ -128,9 +143,20 @@ \begin{figure}[ht] \caption{3D FichCube adaptiv anisotrop vollanalytisch $V\phi = 1$} \centering -\label{fig:exmplAA_3DQuad} +\label{fig:exmplAA_3DQuad_A} \subfloat[Fehler]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_3DFichCube_error}} \subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_3DFichCube_hminmax}}\\ \subfloat[Kondition]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_3DFichCube_cond}} \end{figure} + +\begin{figure}[ht] +\caption{2D Quad adaptiv anisotrop im Vergleich $V\phi = 1$} +\centering +\label{fig:exmplAA_2DQuad_QEQA} +\subfloat[Fehler]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_2DQuad_error}} +\subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_2DQuad_hminmax}}\\ +\subfloat[Kondition]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_2DQuad_cond}} + \subfloat[Zeit]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_2DQuad_time}} +\end{figure} + \end{document} diff --git a/doc/doc.pdf b/doc/doc.pdf index 8ce81b500b2eed8a0830731a78f1343c5012f721..9ba726ca60d5725cecbf0b763200f20c3e0900dc 100644 GIT binary patch delta 325774 zcmV(pK=8k@rW?Vr8<0qU%W@Sr5Z&LexXYPTjUu&LuO%dqA{D4WT-i`rfE{cXwgDT+ zW0POc>3KH@9$u+)$5>A(8jEHUaf}VX3fDV9oNAn6=}VHbG&-q*p`*H#)L+- zopq#`n|p0XDn;vj^R^}I=wfP4+b$>&T+j(Wkb900_HYkfpG%uv*mYG}gHoxqA zzk0Bo zTBCfR=Dj>MtV45uh65;L3{&&5rKoJ;)ch)sqwzS^sZ7z%;41!|v&s|`-iebs! zmb}v1#CdBhGzx^p=W%&MxryPBMkVJ+4v94=YUu2T)y`MXaeo~Srr5{kLEEtk!D(3w zR$v1KhYbj;I>OPUFH-W54id+}Tn>CM(>=#T&4+2uJb7P#?$Bthy%V9BF%);bvxobo 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$T_3$} +\psfrag{T4}{\scriptsize $T_4$} + +\psfrag{x}{\scriptsize $x$} +\psfrag{y}{\scriptsize $y$} +\psfrag{z}{\scriptsize $z$} + \begin{document} \todo{ % \input{titelseite} @@ -138,8 +149,8 @@ In Abschnitt 3 werden wir uns mit der approximativen Berechnung des Doppelintegr \begin{align}\label{math:intro:int} \int_{T_j} \int_{T_k} \kappa(\bs x, \bs y) ds_{\bs y} ds_{\bs x} \end{align} -beschäftigen. $\kappa(\bs x, \bs y)$ sei hierbei eine asymptotisch glatte Kernfunktion. Speziell interessieren wir uns hierbei für die asymptotisch glatte Funktion $\abs{\bs x - \bs y}^{-1}$. Ziel wird es sein, das äußere Integral durch eine Gauss-Quadratur zu erstetzen. Unter bestimmten Zulässigkeits\-beding\-ungen werden wir zeigen, dass die Energienorm durch die Gauss-Quadratur bei steigendem Quadraturgrad exponentiell schnell gegen den exakten Wert konvergiert. Weiterhin betrachten wir die in Abschnitt 2 auftretende Matrix $A \in \R^{n\times n}$, deren Einträge $A_{jk}$ durch \eqref{math:intro:int} bestimmt werden. Denn wir werden eine approximative Matrix $A_p$ aufstellen, welche das durch Quadratur approximierte Integral für die zulässigen Einträge und für alle anderen das exakte Integral verwendet. Hiermit können wir dann Zeigen, dass die approximative Matrix unter der Frobeninusnorm exponentiell schnell gegen $A$ konvergiert.\\ -In Abschnitt 4 fassen wir kurz zusammen, wie wir das Doppelintegral in einfache Integrale zerlegen und andschließend voll analytisch berechnen können. Hierzu werden wir uns weitgehend an \cite{mai:3dbem} orientieren.\\ +beschäftigen. $\kappa(\bs x, \bs y)$ sei hierbei eine asymptotisch glatte Kernfunktion. Speziell interessieren wir uns hierbei für die asymptotisch glatte Funktion $\abs{\bs x - \bs y}^{-1}$. Ziel wird es sein, das äußere Integral durch eine Gauss-Quadratur zu ersetzen. Unter bestimmten Zulässigkeits\-beding\-ungen werden wir zeigen, dass die Energienorm durch die Gauss-Quadratur bei steigendem Quadraturgrad exponentiell schnell gegen den exakten Wert konvergiert. Weiterhin betrachten wir die in Abschnitt 2 auftretende Matrix $A \in \R^{n\times n}$, deren Einträge $A_{jk}$ durch \eqref{math:intro:int} bestimmt werden. Denn wir werden eine approximative Matrix $A_p$ aufstellen, welche das durch Quadratur approximierte Integral für die zulässigen Einträge und für alle anderen das exakte Integral verwendet. Hiermit können wir dann Zeigen, dass die approximative Matrix unter der Frobeninusnorm exponentiell schnell gegen $A$ konvergiert.\\ +In Abschnitt 4 fassen wir kurz zusammen, wie wir das Doppelintegral in einfache Integrale zerlegen und anschließend voll analytisch berechnen können. Hierzu werden wir uns weitgehend an \cite{mai:3dbem} orientieren.\\ Abschließend werden wir kurz die numerische Umsetzung der Techniken vorstellen und anhand von numerischen Beispielen vergleichen. Hierbei wird uns die voll analytische und approximative Berechnung, sowie die adaptive und uniforme Netzverfeinerung besonders interessieren. \clearpage @@ -310,7 +321,7 @@ Des Weiteren werden wir für die Berechnungen noch Aussagen über die Größe ei \begin{align*} \diam (T) &= (a^2+b^2)^{1/2} \end{align*} -Durchmesser von T. Weiterhin nennen wir +Durchmesser von $T$. Weiterhin nennen wir \begin{align*} \diam_{\bs a} (T) = a \end{align*} @@ -405,14 +416,14 @@ Ziel dieses Abschnitts ist die approximative Berechnung des Integrals \begin{align}\label{math:gal:kap} A_{jk} &= \int_{T_j} \int_{T_k} \kappa(\bs x,\bs y) ds_{\bs y} ds_{\bs x}, \end{align} -beziehungsweise als Spezialfall davon die Berechnung des Integrals +auf achsenorientierten Rechtecken $T_j,T_k \subset \R^3$ beziehungsweise als Spezialfall davon die Berechnung des Integrals \begin{align}\label{math:gal:kap+} A_{jk} &= \int_{T_j} \int_{T_k} \frac{1}{|\bs x- \bs y|} ds_{\bs y} ds_{\bs x}. \end{align} unter bestimmten Voraussetzungen an die affinen Randstücke $T_j,T_k$ und den asymptotisch glatten Integranden $\kappa : \R^3 \times \R^3 \to \R$. \subsection{Interpolation} -An dieser stelle werden wir zunächst den Interpolationsoperator auf dem Intervall $[0,1]$ definieren. Ferner wollen wir mithilfe von Chebyshev-Knoten einen Fehlerschätzer für die Chebyshev'sche Interpolation auf Intervallen $[0,1]^d$ mit $d \in \N$ definieren. Im Folgenden bezeichnet $\P^p$ die Menge aller Polynome vom Grad $\leq p$ auf $[0,1]$. +An dieser Stelle werden wir zunächst den Interpolationsoperator auf dem Intervall $[0,1]$ definieren. Ferner wollen wir mithilfe von Chebyshev-Knoten einen Fehlerschätzer für die Chebyshev'sche Interpolation auf Intervallen $[0,1]^d$ mit $d \in \N$ definieren. Im Folgenden bezeichnet $\P^p$ die Menge aller Polynome vom Grad $\leq p$ auf $[0,1]$. \begin{defi} Für einen festen Grad $p \in \N$ und paarweise verschiedene Knoten $x_j \in [0,1]$ lautet das Lagrange'sche Interpolationsproblem: @@ -804,7 +815,7 @@ Wir wollen nun zeigen, dass die approximierende Matrix bezüglich der Frobenius- \end{sat} -\begin{beweis} Betrachten wir zunächst die Differenz zwischen $A$ und $A_p$ in einem festen Eintrag $(A-A_p)_{jk}$. Sind $T_j$ und $T_k$ unzulässig, ist die Differenz laut Definition $0$ und die Abschätzung für $\tilde C_{\zeta_Q,j,k} = 0$ erfüllt. Sind $T_j$ und $T_k$ hingegen zulässig, können wir Satz \ref{thm:sem:quad:V} anwenden. Damit erhalten wir +\begin{beweis} Betrachten wir zunächst die Differenz zwischen $A$ und $A_p$ in einem festen Eintrag $(A-A_p)_{jk}$. Sind $T_j$ und $T_k$ unzulässig, ist die Differenz laut Definition $0$ und die Abschätzung mit $\tilde C_{\zeta_Q,j,k} = 0$ erfüllt. Sind $T_j$ und $T_k$ hingegen zulässig, können wir Satz \ref{thm:sem:quad:V} anwenden. Damit erhalten wir \begin{align*} \norm{A-A_p}_F^2 &= \sum_{j,k=1}^n (A_{jk} - (A_p)_{jk})^2\\ &\leq \sum_{j,k=1}^n \left(\tilde C_{\zeta_Q,j,k}\Lambda_{2p+1}^4 2(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}\right)^2\\ @@ -987,7 +998,7 @@ weshalb die Behauptung folgt. \end{sat} -\begin{beweis} Betrachten wir zunächst die Differenz zwischen $A$ und $A_p$ in einem festen Eintrag $(A-A_p)_{jk}$. Sind $T_j$ und $T_k$ unzulässig, ist die Differenz laut Definition $0$ und die Abschätzung für $\tilde C_{\zeta_E,j,k} = 0$ erfüllt. Sind $T_j$ und $T_k$ hingegen zulässig, unterscheiden wir zwei Fälle. Ist $\diam(T_j) \leq \diam (T_k)$, können wir Satz \ref{thm:sem:quad:E} anwenden. Andernfalls ist durch Lemma \ref{thm:sem:switch} $A_{jk}=A_{kj}$, worauf wir dann Satz \ref{thm:sem:quad:E} anwenden können und dadurch die selbe Abschätzung erhalten. Daraus folgt +\begin{beweis} Betrachten wir zunächst die Differenz zwischen $A$ und $A_p$ in einem festen Eintrag $(A-A_p)_{jk}$. Sind $T_j$ und $T_k$ unzulässig, ist die Differenz laut Definition $0$ und die Abschätzung mit $\tilde C_{\zeta_E,j,k} = 0$ erfüllt. Sind $T_j$ und $T_k$ hingegen zulässig, unterscheiden wir zwei Fälle. Ist $\diam(T_j) \leq \diam (T_k)$, können wir Satz \ref{thm:sem:quad:E} anwenden. Andernfalls ist durch Lemma \ref{thm:sem:switch} $A_{jk}=A_{kj}$, worauf wir dann Satz \ref{thm:sem:quad:E} anwenden können und dadurch die selbe Abschätzung erhalten. Daraus folgt \begin{align*} \norm{A-A_p}_F^2 &= \sum_{j,k=1}^n (A_{jk} - (A_p)_{jk})^2\\ &\leq \sum_{j,k=1}^n \left(\tilde C_{\zeta_E,j,k}\Lambda_{2p+1}^2 2(p+1)\left(1+\sqrt 2 c_2\zeta_E\right)^{-2(p+1)}\right)^2\\ @@ -1353,7 +1364,7 @@ In diesem Abschnitt wollen wir uns mit der analytischen Berechnung des Integrals \begin{align}\label{math:analy:int} A_{jk} = \int_{T_j} \int_{T_k} \frac{1}{|\bs x- \bs y|} ds_{\bs y} ds_{\bs x} \in \R^3. \end{align} -mit zwei beschränkten, achsenorientierten Rechtecken $T_j,T_k \subseteq\R^3$ beschäftigen. Die im Folgenden auftretenden Stammfunktionen $\int f(x) dx$ werden wir der Einfachheit halber jeweils mit additiver Verschiebung $0$ schreiben. +auf zwei beschränkten, achsenorientierten Rechtecken $T_j,T_k \subseteq\R^3$ beschäftigen. Die im Folgenden auftretenden Stammfunktionen $\int f(x) dx$ werden wir der Einfachheit halber jeweils mit additiver Verschiebung $0$ schreiben. Dazu wollen wir \cite{mai:3dbem} folgend, zwei Stammfunktionen zitieren, welche durch das Aufspalten des Integrals \eqref{math:analy:int} auftreten werden. \\\noindent \begin{lem} diff --git a/doc/fig/132t05n05_2DQuad_cond.eps b/doc/fig/132t05n05_2DQuad_cond.eps new file mode 100644 index 0000000..ac68ced --- /dev/null +++ b/doc/fig/132t05n05_2DQuad_cond.eps @@ -0,0 +1,878 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%Creator: MATLAB, The MathWorks, Inc. Version 7.14.0.739 (R2012a). 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+/pgsv () def +/bpage {/pgsv save def} bdef +/epage {pgsv restore} bdef +/bplot /gsave ldef +/eplot {stroke grestore} bdef +% orientation switch +/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def +% coordinate system mappings +/dpi2point 0 def +% font control +/FontSize 0 def +/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] + makefont setfont} bdef +/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse + exch dup 3 1 roll findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def currentdict end definefont pop} bdef +/isroman {findfont /CharStrings get /Agrave known} bdef +/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse + exch FMS} bdef +/csm {1 dpi2point div -1 dpi2point div scale neg translate + dup landscapeMode eq {pop -90 rotate} + {rotateMode eq {90 rotate} if} ifelse} bdef +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef +/DA { [6 dpi2point mul] 0 setdash } bdef +/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 + dpi2point mul] 0 setdash } bdef +% macros for lines and objects +/L {lineto stroke} bdef +/MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef +/AP {{rlineto} repeat} bdef +/PDlw -1 def +/W {/PDlw currentlinewidth def setlinewidth} def +/PP {closepath eofill} bdef +/DP {closepath stroke} bdef +/MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto + neg 0 exch rlineto closepath} bdef +/FR {MR stroke} bdef +/PR {MR fill} bdef +/L1i {{currentfile picstr readhexstring pop} image} bdef +/tMatrix matrix def +/MakeOval {newpath tMatrix currentmatrix pop translate scale +0 0 1 0 360 arc tMatrix setmatrix} bdef +/FO {MakeOval stroke} bdef +/PO {MakeOval fill} bdef +/PD {currentlinewidth 2 div 0 360 arc fill + PDlw -1 eq not {PDlw w /PDlw -1 def} if} def +/FA {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef +/PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef +/FAn {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef +/PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef +/vradius 0 def /hradius 0 def /lry 0 def +/lrx 0 def /uly 0 def /ulx 0 def /rad 0 def +/MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef + /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly + vradius add translate hradius vradius scale 0 0 1 180 270 arc + tMatrix setmatrix lrx hradius sub uly vradius add translate + hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix + lrx hradius sub lry vradius sub translate hradius vradius scale + 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub + translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix + closepath} bdef +/FRR {MRR stroke } bdef +/PRR {MRR fill } bdef +/MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix + closepath} bdef +/FlrRR {MlrRR stroke } bdef +/PlrRR {MlrRR fill } bdef +/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix + closepath} bdef +/FtbRR {MtbRR stroke } bdef +/PtbRR {MtbRR fill } bdef +/stri 6 array def /dtri 6 array def +/smat 6 array def /dmat 6 array def +/tmat1 6 array def /tmat2 6 array def /dif 3 array def +/asub {/ind2 exch def /ind1 exch def dup dup + ind1 get exch ind2 get sub exch } bdef +/tri_to_matrix { + 2 0 asub 3 1 asub 4 0 asub 5 1 asub + dup 0 get exch 1 get 7 -1 roll astore } bdef +/compute_transform { + dmat dtri tri_to_matrix tmat1 invertmatrix + smat stri tri_to_matrix tmat2 concatmatrix } bdef +/ds {stri astore pop} bdef +/dt {dtri astore pop} bdef +/db {2 copy /cols xdef /rows xdef mul dup 3 mul string + currentfile exch readhexstring pop + dup 0 3 index getinterval /rbmap xdef + dup 2 index dup getinterval /gbmap xdef + 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef +/it {gs np dtri aload pop moveto lineto lineto cp c + cols rows 8 compute_transform + rbmap gbmap bbmap true 3 colorimage gr}bdef +/il {newpath moveto lineto stroke}bdef +currentdict end def +%%EndProlog + +%%BeginSetup +MathWorks begin + +0 cap + +end +%%EndSetup + +%%Page: 1 1 +%%BeginPageSetup +%%PageBoundingBox: 53 195 550 592 +MathWorks begin +bpage +%%EndPageSetup + +%%BeginObject: obj1 +bplot + +/dpi2point 8.33333 def +portraitMode 0150 5100 csm + + 294 165 4145 3302 MR c np +83 dict begin %Colortable 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+/pgsv () def +/bpage {/pgsv save def} bdef +/epage {pgsv restore} bdef +/bplot /gsave ldef +/eplot {stroke grestore} bdef +% orientation switch +/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def +% coordinate system mappings +/dpi2point 0 def +% font control +/FontSize 0 def +/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] + makefont setfont} bdef +/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse + exch dup 3 1 roll findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def currentdict end definefont pop} bdef +/isroman {findfont /CharStrings get /Agrave known} bdef +/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse + exch FMS} bdef +/csm {1 dpi2point div -1 dpi2point div scale neg translate + dup landscapeMode eq {pop -90 rotate} + {rotateMode eq {90 rotate} if} ifelse} bdef +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef +/DA { [6 dpi2point mul] 0 setdash } bdef +/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 + dpi2point mul] 0 setdash } bdef +% macros for lines and objects +/L {lineto stroke} bdef +/MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef +/AP {{rlineto} repeat} bdef +/PDlw -1 def +/W {/PDlw currentlinewidth def setlinewidth} def +/PP {closepath eofill} bdef +/DP {closepath stroke} bdef +/MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto + neg 0 exch rlineto closepath} bdef +/FR {MR stroke} bdef +/PR {MR fill} bdef +/L1i {{currentfile picstr readhexstring pop} image} bdef +/tMatrix matrix def +/MakeOval {newpath tMatrix currentmatrix pop translate scale +0 0 1 0 360 arc tMatrix setmatrix} bdef +/FO {MakeOval stroke} bdef +/PO {MakeOval fill} bdef +/PD {currentlinewidth 2 div 0 360 arc fill + PDlw -1 eq not {PDlw w /PDlw -1 def} if} def +/FA {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef +/PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef +/FAn {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef +/PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef +/vradius 0 def /hradius 0 def /lry 0 def +/lrx 0 def /uly 0 def /ulx 0 def /rad 0 def +/MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef + /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly + vradius add translate hradius vradius scale 0 0 1 180 270 arc + tMatrix setmatrix lrx hradius sub uly vradius add translate + hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix + lrx hradius sub lry vradius sub translate hradius vradius scale + 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub + translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix + closepath} bdef +/FRR {MRR stroke } bdef +/PRR {MRR fill } bdef +/MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix + closepath} bdef +/FlrRR {MlrRR stroke } bdef +/PlrRR {MlrRR fill } bdef +/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix + closepath} bdef +/FtbRR {MtbRR stroke } bdef +/PtbRR {MtbRR fill } bdef +/stri 6 array def /dtri 6 array def +/smat 6 array def /dmat 6 array def +/tmat1 6 array def /tmat2 6 array def /dif 3 array def +/asub {/ind2 exch def /ind1 exch def dup dup + ind1 get exch ind2 get sub exch } bdef +/tri_to_matrix { + 2 0 asub 3 1 asub 4 0 asub 5 1 asub + dup 0 get exch 1 get 7 -1 roll astore } bdef +/compute_transform { + dmat dtri tri_to_matrix tmat1 invertmatrix + smat stri tri_to_matrix tmat2 concatmatrix } bdef +/ds {stri astore pop} bdef +/dt {dtri astore pop} bdef +/db {2 copy /cols xdef /rows xdef mul dup 3 mul string + currentfile exch readhexstring pop + dup 0 3 index getinterval /rbmap xdef + dup 2 index dup getinterval /gbmap xdef + 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef +/it {gs np dtri aload pop moveto lineto lineto cp c + cols rows 8 compute_transform + rbmap gbmap bbmap true 3 colorimage gr}bdef +/il {newpath moveto lineto stroke}bdef +currentdict end def +%%EndProlog + +%%BeginSetup +MathWorks begin + +0 cap + +end +%%EndSetup + +%%Page: 1 1 +%%BeginPageSetup +%%PageBoundingBox: 53 195 550 592 +MathWorks begin +bpage +%%EndPageSetup + +%%BeginObject: obj1 +bplot + +/dpi2point 8.33333 def +portraitMode 0150 5100 csm + + 294 165 4145 3302 MR c np +82 dict begin %Colortable 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b/doc/fig/132t05n05_2DQuad_time.eps @@ -0,0 +1,852 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%Creator: MATLAB, The MathWorks, Inc. Version 7.14.0.739 (R2012a). Operating System: Linux 3.2.0-40-generic #64-Ubuntu SMP Mon Mar 25 21:22:10 UTC 2013 x86_64. +%%Title: ../doc/fig/132t05n05_2DQuad_time.eps +%%CreationDate: 04/03/2013 12:25:13 +%%DocumentNeededFonts: Helvetica +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%Extensions: CMYK +%%Pages: 1 +%%BoundingBox: 53 195 550 592 +%%EndComments + +%%BeginProlog +% MathWorks dictionary +/MathWorks 160 dict begin +% definition operators +/bdef {bind def} bind def +/ldef {load def} bind def +/xdef {exch def} bdef +/xstore {exch store} bdef +% operator abbreviations +/c /clip ldef +/cc /concat ldef +/cp /closepath ldef +/gr /grestore ldef +/gs /gsave ldef +/mt /moveto ldef +/np /newpath ldef +/cm /currentmatrix ldef +/sm /setmatrix ldef +/rm /rmoveto ldef +/rl /rlineto ldef +/s {show newpath} bdef +/sc {setcmykcolor} bdef +/sr /setrgbcolor ldef +/sg /setgray ldef +/w /setlinewidth ldef +/j /setlinejoin ldef +/cap /setlinecap ldef +/rc {rectclip} bdef +/rf {rectfill} bdef +% page state control +/pgsv () def +/bpage {/pgsv save def} bdef +/epage {pgsv restore} bdef +/bplot /gsave ldef +/eplot {stroke grestore} bdef +% orientation switch +/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def +% coordinate system mappings +/dpi2point 0 def +% font control +/FontSize 0 def +/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] + makefont setfont} bdef +/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse + exch dup 3 1 roll findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def currentdict end definefont pop} bdef +/isroman {findfont /CharStrings get /Agrave known} bdef +/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse + exch FMS} bdef +/csm {1 dpi2point div -1 dpi2point div scale neg translate + dup landscapeMode eq {pop -90 rotate} + {rotateMode eq {90 rotate} if} ifelse} bdef +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef +/DA { [6 dpi2point mul] 0 setdash } bdef +/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 + dpi2point mul] 0 setdash } bdef +% macros for lines and objects +/L {lineto stroke} bdef +/MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef +/AP {{rlineto} repeat} bdef +/PDlw -1 def +/W {/PDlw currentlinewidth def setlinewidth} def +/PP {closepath eofill} bdef +/DP {closepath stroke} bdef +/MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto + neg 0 exch rlineto closepath} bdef +/FR {MR stroke} bdef +/PR {MR fill} bdef +/L1i {{currentfile picstr readhexstring pop} image} bdef +/tMatrix matrix def +/MakeOval {newpath tMatrix currentmatrix pop translate scale +0 0 1 0 360 arc tMatrix setmatrix} bdef +/FO {MakeOval stroke} bdef +/PO {MakeOval fill} bdef +/PD {currentlinewidth 2 div 0 360 arc fill + PDlw -1 eq not {PDlw w /PDlw -1 def} if} def +/FA {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef +/PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef +/FAn {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef +/PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef +/vradius 0 def /hradius 0 def /lry 0 def +/lrx 0 def /uly 0 def /ulx 0 def /rad 0 def +/MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef + /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly + vradius add translate hradius vradius scale 0 0 1 180 270 arc + tMatrix setmatrix lrx hradius sub uly vradius add translate + hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix + lrx hradius sub lry vradius sub translate hradius vradius scale + 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub + translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix + closepath} bdef +/FRR {MRR stroke } bdef +/PRR {MRR fill } bdef +/MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix + closepath} bdef +/FlrRR {MlrRR stroke } bdef +/PlrRR {MlrRR fill } bdef +/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix + closepath} bdef +/FtbRR {MtbRR stroke } bdef +/PtbRR {MtbRR fill } bdef +/stri 6 array def /dtri 6 array def +/smat 6 array def /dmat 6 array def +/tmat1 6 array def /tmat2 6 array def /dif 3 array def +/asub {/ind2 exch def /ind1 exch def dup dup + ind1 get exch ind2 get sub exch } bdef +/tri_to_matrix { + 2 0 asub 3 1 asub 4 0 asub 5 1 asub + dup 0 get exch 1 get 7 -1 roll astore } bdef +/compute_transform { + dmat dtri tri_to_matrix tmat1 invertmatrix + smat stri tri_to_matrix tmat2 concatmatrix } bdef +/ds {stri astore pop} bdef +/dt {dtri astore pop} bdef +/db {2 copy /cols xdef /rows xdef mul dup 3 mul string + currentfile exch readhexstring pop + dup 0 3 index getinterval /rbmap xdef + dup 2 index dup getinterval /gbmap xdef + 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef +/it {gs np dtri aload pop moveto lineto lineto cp c + cols rows 8 compute_transform + rbmap gbmap bbmap true 3 colorimage gr}bdef +/il {newpath moveto lineto stroke}bdef +currentdict end def +%%EndProlog + +%%BeginSetup +MathWorks begin + +0 cap + +end +%%EndSetup + +%%Page: 1 1 +%%BeginPageSetup +%%PageBoundingBox: 53 195 550 592 +MathWorks begin +bpage +%%EndPageSetup + +%%BeginObject: obj1 +bplot + +/dpi2point 8.33333 def +portraitMode 0150 5100 csm + + 294 165 4145 3302 MR c np +82 dict begin %Colortable 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Operating System: Linux 3.2.0-40-generic #64-Ubuntu SMP Mon Mar 25 21:22:10 UTC 2013 x86_64. %%Title: ../doc/fig/exmpl12_ref.eps -%%CreationDate: 03/30/2013 22:56:58 +%%CreationDate: 04/03/2013 12:25:00 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK @@ -170,7 +170,7 @@ bplot portraitMode 0150 5100 csm 334 236 4045 3206 MR c np -84 dict begin %Colortable dictionary +86 dict begin %Colortable dictionary /c0 { 0.000000 0.000000 0.000000 sr} bdef /c1 { 1.000000 1.000000 1.000000 sr} bdef /c2 { 0.900000 0.000000 0.000000 sr} bdef diff --git a/doc/fig/exmpl13_ref.eps b/doc/fig/exmpl13_ref.eps index 1a46389..a79fba3 100644 --- a/doc/fig/exmpl13_ref.eps +++ b/doc/fig/exmpl13_ref.eps @@ -1,7 +1,7 @@ %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The MathWorks, Inc. Version 7.14.0.739 (R2012a). Operating System: Linux 3.2.0-40-generic #64-Ubuntu SMP Mon Mar 25 21:22:10 UTC 2013 x86_64. %%Title: ../doc/fig/exmpl13_ref.eps -%%CreationDate: 03/30/2013 22:56:58 +%%CreationDate: 04/03/2013 12:25:00 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK @@ -170,7 +170,7 @@ bplot portraitMode 0150 5100 csm 334 236 4045 3206 MR c np -84 dict begin %Colortable dictionary +86 dict begin %Colortable dictionary /c0 { 0.000000 0.000000 0.000000 sr} bdef /c1 { 1.000000 1.000000 1.000000 sr} bdef /c2 { 0.900000 0.000000 0.000000 sr} bdef diff --git a/doc/fig/exmpl_2DLShape_ref.eps b/doc/fig/exmpl_2DLShape_ref.eps index 43ae4c1..a67c28f 100644 --- a/doc/fig/exmpl_2DLShape_ref.eps +++ b/doc/fig/exmpl_2DLShape_ref.eps @@ -1,7 +1,7 @@ %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The MathWorks, Inc. Version 7.14.0.739 (R2012a). Operating System: Linux 3.2.0-40-generic #64-Ubuntu SMP Mon Mar 25 21:22:10 UTC 2013 x86_64. %%Title: ../doc/fig/exmpl_2DLShape_ref.eps -%%CreationDate: 03/30/2013 22:56:59 +%%CreationDate: 04/03/2013 12:25:01 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK @@ -170,7 +170,7 @@ bplot portraitMode 0150 5100 csm 334 236 4045 3206 MR c np -84 dict begin %Colortable dictionary +86 dict begin %Colortable dictionary /c0 { 0.000000 0.000000 0.000000 sr} bdef /c1 { 1.000000 1.000000 1.000000 sr} bdef /c2 { 0.900000 0.000000 0.000000 sr} bdef diff --git a/doc/fig/exmpl_2DQuad_ref.eps b/doc/fig/exmpl_2DQuad_ref.eps index 52ccc57..d1a2088 100644 --- a/doc/fig/exmpl_2DQuad_ref.eps +++ b/doc/fig/exmpl_2DQuad_ref.eps @@ -1,7 +1,7 @@ %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The MathWorks, Inc. Version 7.14.0.739 (R2012a). Operating System: Linux 3.2.0-40-generic #64-Ubuntu SMP Mon Mar 25 21:22:10 UTC 2013 x86_64. %%Title: ../doc/fig/exmpl_2DQuad_ref.eps -%%CreationDate: 03/30/2013 22:56:59 +%%CreationDate: 04/03/2013 12:25:01 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK @@ -170,7 +170,7 @@ bplot portraitMode 0150 5100 csm 334 236 4045 3206 MR c np -84 dict begin %Colortable dictionary +86 dict begin %Colortable dictionary /c0 { 0.000000 0.000000 0.000000 sr} bdef /c1 { 1.000000 1.000000 1.000000 sr} bdef /c2 { 0.900000 0.000000 0.000000 sr} bdef diff --git a/doc/fig/exmpl_3DCube_ref.eps b/doc/fig/exmpl_3DCube_ref.eps index 81e8c59..a6d0cb7 100644 --- a/doc/fig/exmpl_3DCube_ref.eps +++ b/doc/fig/exmpl_3DCube_ref.eps @@ -1,7 +1,7 @@ %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The MathWorks, Inc. Version 7.14.0.739 (R2012a). Operating System: Linux 3.2.0-40-generic #64-Ubuntu SMP Mon Mar 25 21:22:10 UTC 2013 x86_64. %%Title: ../doc/fig/exmpl_3DCube_ref.eps -%%CreationDate: 03/30/2013 22:56:59 +%%CreationDate: 04/03/2013 12:25:01 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK @@ -170,7 +170,7 @@ bplot portraitMode 0150 5100 csm 286 236 4092 3188 MR c np -85 dict begin %Colortable dictionary +87 dict begin %Colortable dictionary /c0 { 0.000000 0.000000 0.000000 sr} bdef /c1 { 1.000000 1.000000 1.000000 sr} bdef /c2 { 0.900000 0.000000 0.000000 sr} bdef diff --git a/doc/fig/exmpl_3DFichCube_ref.eps b/doc/fig/exmpl_3DFichCube_ref.eps index dd42f83..518c758 100644 --- a/doc/fig/exmpl_3DFichCube_ref.eps +++ b/doc/fig/exmpl_3DFichCube_ref.eps @@ -1,7 +1,7 @@ %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The MathWorks, Inc. Version 7.14.0.739 (R2012a). Operating System: Linux 3.2.0-40-generic #64-Ubuntu SMP Mon Mar 25 21:22:10 UTC 2013 x86_64. %%Title: ../doc/fig/exmpl_3DFichCube_ref.eps -%%CreationDate: 03/30/2013 22:56:59 +%%CreationDate: 04/03/2013 12:25:02 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK @@ -170,7 +170,7 @@ bplot portraitMode 0150 5100 csm 237 236 4142 3188 MR c np -85 dict begin %Colortable dictionary +87 dict begin %Colortable dictionary /c0 { 0.000000 0.000000 0.000000 sr} bdef /c1 { 1.000000 1.000000 1.000000 sr} bdef /c2 { 0.900000 0.000000 0.000000 sr} bdef diff --git a/doc/fig/net1.eps b/doc/fig/net1.eps index 45bb754..34e4c5b 100644 --- a/doc/fig/net1.eps +++ b/doc/fig/net1.eps @@ -1,7 +1,7 @@ %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The MathWorks, Inc. Version 7.14.0.739 (R2012a). Operating System: Linux 3.2.0-40-generic #64-Ubuntu SMP Mon Mar 25 21:22:10 UTC 2013 x86_64. %%Title: ../doc/fig/net1.eps -%%CreationDate: 03/30/2013 22:56:58 +%%CreationDate: 04/03/2013 12:25:01 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK @@ -170,7 +170,7 @@ bplot portraitMode 0150 5100 csm 916 463 3169 2517 MR c np -83 dict begin %Colortable dictionary +86 dict begin %Colortable dictionary /c0 { 0.000000 0.000000 0.000000 sr} bdef /c1 { 1.000000 1.000000 1.000000 sr} bdef /c2 { 0.900000 0.000000 0.000000 sr} bdef diff --git a/doc/fig/net_single.eps b/doc/fig/net_single.eps index ec690ba..d448dfa 100644 --- a/doc/fig/net_single.eps +++ b/doc/fig/net_single.eps @@ -1,7 +1,7 @@ %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The MathWorks, Inc. Version 7.14.0.739 (R2012a). Operating System: Linux 3.2.0-40-generic #64-Ubuntu SMP Mon Mar 25 21:22:10 UTC 2013 x86_64. %%Title: ../doc/fig/net_single.eps -%%CreationDate: 03/30/2013 22:56:59 +%%CreationDate: 04/03/2013 12:25:01 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK @@ -170,7 +170,7 @@ bplot portraitMode 0150 5100 csm 286 236 4093 3188 MR c np -84 dict begin %Colortable dictionary +86 dict begin %Colortable dictionary /c0 { 0.000000 0.000000 0.000000 sr} bdef /c1 { 1.000000 1.000000 1.000000 sr} bdef /c2 { 0.900000 0.000000 0.000000 sr} bdef @@ -258,6 +258,8 @@ SO 623 840 mt 574 817 L 506 837 mt (1) s +4121 2627 mt +(k2) s gs 624 269 3721 2937 MR c np 16.6667 w /c8 { 0.000000 0.400000 0.400000 sr} bdef @@ -268,31 +270,36 @@ gr 16.6667 w c8 0 sg +3448 2317 mt +(e2) s +3224 2919 mt +(e1) s 3194 3389 mt (x) s +2708 1977 mt +(k3) s gs 624 269 3721 2937 MR c np c8 2688 2001 mt 933 2477 L +gr + +c8 +0 sg +2571 1419 mt +(n) s +gs 624 269 3721 2937 MR c np 4.16667 w /c9 { 1.000000 0.000000 0.000000 sr} bdef c9 2484 2549 mt 2484 926 L +0 j gr 4.16667 w c9 - 25 25 2484 2549 FO -gs 624 269 3721 2937 MR c np -0 j -gr - 2467 909 mt 2501 943 L 2501 909 mt 2467 943 L gs 624 269 3721 2937 MR c np -gr - - 25 25 2484 926 FO -gs 624 269 3721 2937 MR c np 16.6667 w c8 2279 3096 mt 4034 2621 L @@ -301,6 +308,12 @@ gr 16.6667 w c8 0 sg +2259 3181 mt +(k1) s +1764 2183 mt +(e3) s +1519 2840 mt +(e4) s 1363 3346 mt (y) s gs 624 269 3721 2937 MR c np @@ -310,6 +323,8 @@ gr c8 0 sg + 778 2500 mt +(k4) s 380 1673 mt -90 rotate (z) s 90 rotate diff --git a/doc/fig/net_wrong.eps b/doc/fig/net_wrong.eps index 8e097d8..0a3f740 100644 --- a/doc/fig/net_wrong.eps +++ b/doc/fig/net_wrong.eps @@ -1,7 +1,7 @@ %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The MathWorks, Inc. Version 7.14.0.739 (R2012a). Operating System: Linux 3.2.0-40-generic #64-Ubuntu SMP Mon Mar 25 21:22:10 UTC 2013 x86_64. %%Title: ../doc/fig/net_wrong.eps -%%CreationDate: 03/30/2013 22:56:58 +%%CreationDate: 04/03/2013 12:25:01 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK @@ -170,7 +170,7 @@ bplot portraitMode 0150 5100 csm 916 463 3169 2517 MR c np -83 dict begin %Colortable dictionary +86 dict begin %Colortable dictionary /c0 { 0.000000 0.000000 0.000000 sr} bdef /c1 { 1.000000 1.000000 1.000000 sr} bdef /c2 { 0.900000 0.000000 0.000000 sr} bdef diff --git a/src/A_plots.m b/src/A_plots.m index 83128ab..b7b358b 100644 --- a/src/A_plots.m +++ b/src/A_plots.m @@ -92,7 +92,7 @@ for i = 1:length(files) }'; leg3 = {leg3{:}... - ['cond(A_{h/2}) ' l0 l1{i}]... + ['cond ' l0 l1{i}]... }'; leg4 = {leg4{:}... ['Zeit ' l0 l1{i}]... diff --git a/src/areaQuad.m b/src/areaQuad.m index 0ccdaca..1f9652d 100644 --- a/src/areaQuad.m +++ b/src/areaQuad.m @@ -2,7 +2,7 @@ function [area, sites] = areaQuad(sites) % % [area, vector, sites] = area(coordinates, elements) % -% Diese Funktion Berechnet den Flächeninhalt zu jedem Element (area) und +% Diese Funktion Berechnet den Flaecheninhalt zu jedem Element (area) und % zu jedem Element die Orthogonalen mit Laenge 1, des Weiteren werden auch % die Seitenlaengen zu jedem Element gespeichert % diff --git a/src/compute.m b/src/compute.m index 5c8d2b1..8e67a80 100644 --- a/src/compute.m +++ b/src/compute.m @@ -19,32 +19,31 @@ function [data er fileo] = compute(file,times,zeta,typ,theta,nu,vcon) % Datei laden load(file) +% Kontainer fuer Messungen initialisieren if(~exist('data','var')) data=[]; end + kap3 = 0; -%times mal verfeinern + +%times mal verfeinern oder bis Elementanzahl fuer times > 40 for j = 1:times %Beende Schleife wenn Element Anzahl erreicht if(times>40 && size(elements,1) > times) break; end + + %altes Netz mit Loesung merken if(exist('coo_fine','var')) old_C_fine = coo_fine; old_F_fine = f2s; old_E_fine = ele_fine; old_x_fine = x_fine; end - -% figure(10) -% plotShape(coordinates,elements,''); - + %uniformIsotrop Verfeinern [coo_fine,ele_fine,neigh_fine,f2s,sit_fine]... =refineQuad(coordinates,elements,neigh,sites,2); - -% figure(11) -% plotShape(coordinates,elements,''); %Flaecheninhalte Berechnen (rhs) b_fine = areaQuad(sit_fine); @@ -99,20 +98,13 @@ for j = 1:times clear V_fine - % \tilde \mu ( \Pi h -h + L_2 ) + % \tilde \mu ( \Rho h -h + L_2 ) tmu = hmin.* b .* ... sum((x_fine(f2s)'-repmat(sum(x_fine(f2s)',1)/4,4,1)).^2)' /4; - % Export für Valgrind -% z = zeta{i}; -% t = typ(i); -% save('input.mat','coordinates','elements','z','t'); - - %Fehlerschaetzer 2 aufbauen V = mex_build_V(coordinates,elements,zeta{i},typ(i)); - if(~vcon) x = V\b; else @@ -129,9 +121,7 @@ for j = 1:times xo_fine(f2s) = repmat(x,1,4); xd_fine = abs(xo_fine'-x_fine); - - - % \tilde \mu ( h/2 -h + L_2 ) + % \mu ( h/2 -h + L_2 ) mu = hmin.*b.*sum((x_fine(f2s)'-repmat(x',4,1)).^2)'/4; @@ -143,9 +133,7 @@ for j = 1:times %Enorm^2 Elementweise Vergleich - if(exist('old_F_fine','var')) -% xf_fine = zeros(size(x_fine)); for k = 1:size(ele_fine,1) e_f = find(sum(k==f2s,2)); e_f_p = find(f2s(e_f,:)==k); @@ -165,33 +153,12 @@ for j = 1:times inh = [1 2 4 3]; e_p = inh(e_ff_p(mod(floor((e_f_p)/2),2)+1)); end - end -% figure(11) -% plotShape(old_C,old_E,'b');view(2); -% plotMark(e_ff,coo_fine,ele_fine); -% -% figure(12) -% plotShape(old_C_fine,old_E_fine,'b'); view(2); -% plotMark(old_F_fine(e_ff,e_p),coo_fine,ele_fine); -% -% -% figure(13) -% plotShape(coordinates,elements,'b');view(2); -% plotMark(e_f,coo_fine,ele_fine); -% -% -% figure(14) -% plotShape(coo_fine,ele_fine,'b');view(2); -% plotMark(k,coo_fine,ele_fine); - + end xf_fine(k,1) = abs(x_fine(k) - old_x_fine(old_F_fine(e_ff,e_p))); end kap3 = b_fine'*(xf_fine); end - -% xe_fine = b_fine'*x_fine; - %\tilde \mu 2 = ( ||\Pi h|| - ||h||) @@ -201,15 +168,8 @@ for j = 1:times % |||h/2 -h||| % eta = xd_fine'*A_fine*xd_fine; eta = abs(xe_fine-xe); - -% save_A_fine{i} = V_fine; -% save_x_fine{i} = x_fine; -% -% save_A{i} = V; -% save_x{i} = x; - - -end_time = toc - start_time; + + end_time = toc - start_time; dataS = [dataS ... typ(i) ... berechnet mit Typ @@ -224,15 +184,10 @@ end_time = toc - start_time; sqrt(sum(tmu2))... tilde mu 2 xe_fine... (kappa) kap3 ... kappa 3 - end_time ... benötigte Zeit (Aufbauen, Berechnen) + end_time ... benoetigte Zeit (Aufbauen, Berechnen) ]; end - - % nur RandElemente Verfeinern -% marked = ones(1,size(G_E,1)); -% marked(find(sum((G_N(:,1:4)==0),2))) = 2; - % Markieren mit gewaehlten Parametern marked = mark(x_fine(f2s)',tmu,theta,nu); @@ -251,18 +206,10 @@ end_time = toc - start_time; old_E = elements; old_S = sites; -% figure(1) -% plotShape(coordinates,elements(:,:),'db');view(2); - %Netz Verfeinern, wie durch marked bestimmt [coordinates, elements, neigh, f, sites, er]... = refineQuad(coordinates,elements,neigh,sites,marked); -% figure(2) -% plotShape(coordinates,elements(:,:),'db');view(2); - -% f - %Vater Sohn test assert(sum(areaQuad(old_S))==sum(areaQuad(sites)),... 'Gesamtinhalt Fehlerhaft') @@ -292,8 +239,6 @@ end_time = toc - start_time; else p2(2) = p2(2) -2; end - - %ErgebnisWerte Speichern data(size(data,1)+1,1:length(dataS)) = dataS; @@ -323,16 +268,6 @@ end end -% kappa: -% kappa = ||| phi_{h/2}^{\ell} - \phi_{h/2}^{\ell-1} ||| -% ist die differenz der eNormen von den h/2 Netzen? -% -% tildeMu: -% tilde-mu(T)^2 = hmin(T) || (1-Pi_h) phi_{h/2} ||^2_{L2(T)} -% = hmin(T) * ( ||phi_{h/2}||^2_{L2(T)} - ||Pi_h -% phi_{h/2}||^2_{L2(T)} - - function str = t2str(time) type = 's'; diff --git a/src/export_exmpl.m b/src/export_exmpl.m index fcf238f..68e060f 100644 --- a/src/export_exmpl.m +++ b/src/export_exmpl.m @@ -42,6 +42,21 @@ print('-r600','-depsc',['../doc/fig/net_wrong.eps']) load exmpl_2DQuad2 plotShape(coordinates,elements(:,:),'ben'); +hold on +text(-.05,-.05,0,'k1'); +text(1.05,0,0,'k2'); +text(1.05,1.05,0,'k3'); +text(-.05,1.05,0,'k4'); + +text(.5,-.05,0,'e1'); +text(1.05,.5,0,'e2'); +text(.55,1.1,0,'e3'); +text(-.05,.5,0,'e4'); + +text(.55,.5,.7,'n'); +hold off + + % view(2); % axis off; print('-r600','-depsc',['../doc/fig/net_single.eps']) @@ -73,7 +88,7 @@ A_plots({'meshSave/1t1n0_2DQuad_6',... 'meshSave/1t05n05_2DQuad_29'},'../doc/fig/1tn_2DQuad') %% Semianalytisch -A_plots({'meshSave/132t05n05_2DQuad_29'},'../doc/fig/132t05n05_2DQuad') +A_plots({'meshSave/132t05n05_2DQuad_30'},'../doc/fig/132t05n05_2DQuad') diff --git a/src/refineQuad.m b/src/refineQuad.m index fb64a24..301260d 100644 --- a/src/refineQuad.m +++ b/src/refineQuad.m @@ -3,13 +3,13 @@ function [coo,ele,nei,f2s,sit,err] = refineQuad(coordinates,elements,neigh,sites % [coo,ele,nei,f2s,sit,err] = refineQuad(coordinates,elements,type) % % Verfeinert die markierten Elemente mit dem entsprechenden TYP und gibt -% auch die F2S Beziehungen zurück. type muss von der Länge der Anzahl der -% Elemente entsprechen oder genau 1 und die Einträge können 1,2,3,4,5 sein. +% auch die F2S Beziehungen zurueck. type muss von der Laenge der Anzahl der +% Elemente entsprechen oder genau 1 und die Eintraege koennen 1,2,3,4,5 sein. % % der Typ zu jedem Element entspricht dabei: % 1 - keine Verfeinerung % 2 - 4 neue Elemente -% 3 - 2 neue Elemente, übereinander +% 3 - 2 neue Elemente, uebereinander % 4 - 2 neue Elemente, nebeneinander % 5 - 4 neue Elemente, wird erst (3) und dann beide Elemente (4) geteilt % @@ -34,7 +34,7 @@ global G_ref_S; global G_ref_f2s; %Finale Beziehung (VaterSohn) global G_ref_t; %wie soll verfeinert werden global G_ref_tD; %wie wurde bereits verfeinert (in diesem Durchlauf) -global G_ref_f2sT; %Temporäre Beziehung +global G_ref_f2sT; %Temporaere Beziehung %INTERNE Globale Variablen zuweisen G_ref_E = elements; @@ -42,7 +42,7 @@ G_ref_C = coordinates; G_ref_N = neigh; G_ref_S = sites; G_ref_t = typ; -G_ref_f2s = repmat([1:size(elements,1)]',1,4); +G_ref_f2s = repmat((1:size(elements,1))',1,4); G_ref_tD = ones(size(elements,1),1); %Parameter Freigeben (Speicher...) @@ -63,7 +63,7 @@ while(1==1) t_ref=find(G_ref_t==2); G_ref_t(t_ref(2:4:end)) = 5; - %Welche Elemente müssen bearbeitet werden + %Welche Elemente muessen bearbeitet werden ref = find(G_ref_t>1); ref = reshape(ref,1,length(ref)); @@ -78,22 +78,10 @@ while(1==1) break; end - -% figure(6) -% plotShape(G_ref_C,G_ref_E) -% plotMark(ref,G_ref_C,G_ref_E,'xg') -% title('Zum Verfeinern Markierte Elemente') - % Elementeweise Bearbeiten for ele = ref % ref(randperm(length(ref))) - -% figure(5) -% plotShape(G_ref_C,G_ref_E) -% plotMark(ele,G_ref_C,G_ref_E,'xg') -% view(2) -% title('Entscheidene Nachbarelemente') - - % # HangingNode Check + + % HangingNode Check Nt = find(G_ref_N(ele,5:8)==0); N = G_ref_N(ele,Nt); N2t = find(N~=0); @@ -106,9 +94,7 @@ while(1==1) elseif(G_ref_t(ele)==4) N2 = N2(mod(N2tt,2)==1); end - -% plotMark(N2,G_ref_C,G_ref_E,'or') - + %Hat noch zu teilende Nachbarn? if(~isempty(N2)) N3t = mod(find((G_ref_N(N2',:)==ele)')-1,4)+1; %Nachbarseiten @@ -136,19 +122,18 @@ while(1==1) end - % Da Nachbarn noch Verfeinert werden müssen erst mal weiter + % Da Nachbarn noch Verfeinert werden muessen erst mal weiter continue; end end - % Wenn Alle Überprüfungen durchgelaufen sind + % Wenn Alle Ueberpruefungen durchgelaufen sind assert(G_ref_tD(ele)~=2,'Element ist schon verfeinert') assert(G_ref_t(ele)>1,'Element ist nicht Markiert') G_ref_f2sT = ones(1,4)*ele; refineE(ele); %Element Teilen updateN(ele); %Nachbarn des Elements aktualisieren updateF2S(ele); %VaterSohn Relation setzen -% plotShape(G_ref_C,G_ref_E) end end @@ -160,7 +145,7 @@ f2s = G_ref_f2s; sit = G_ref_S; %Doppelte Koordinaten loeschen -[coo l pos] = unique(coo,'rows'); +[coo , ~, pos] = unique(coo,'rows'); pos = pos'; ele = pos(ele); @@ -168,7 +153,7 @@ ele = pos(ele); clear G_ref_E G_ref_C G_ref_N G_ref_f2s G_ref_t G_ref_tD G_ref_s end -%% Element Verfeinern ! sollte nur ausgeführt werden wenn wirklich möglich +%% Element Verfeinern ! sollte nur ausgefuehrt werden wenn wirklich moeglich function refineE(ele) % Element wird gnadenlos Verfeinert @@ -224,7 +209,7 @@ global G_ref_t; end -%% Aktualisieren der Nachbarn ! sollte nur ausgeführt werden wenn wirklich möglich +%% Aktualisieren der Nachbarn ! sollte nur ausgefuehrt werden wenn wirklich moeglich function updateN(ele) % Nachbarschaften werden neu gesetzt (nach N und f2s) @@ -237,7 +222,6 @@ this = G_ref_N(ele,:); %An welchen Kanten habe ich Nachbarn S = find(mod((this(1:4)~=0).*(this(5:8)==0),2))'; %Einen Nachbar (Single) D = find(this(5:8)~=0)'; %Zwei Nachbarn (Double) -% G_ref_N([this(S) this(D)],:); %An welchen Kanten bin ich Nachbar MSt = mod(find((G_ref_N(this(S),:)==ele)')-1,8)+1; @@ -252,7 +236,7 @@ if(G_ref_t(ele)==3||G_ref_t(ele)==5) % Beziehungen fuer Kanten mit einem Nachbar for i = 1:length(S) - if(mod(S(i),2)==0) %TODO was ist wenn ich selbst doppelNachbar an der Seite bin??? + if(mod(S(i),2)==0) G_ref_N(this(S(i)),[MS(i) MS(i)+4]) = [G_ref_f2sT(S(i)) G_ref_f2sT(mod(S(i),4)+1)]; G_ref_N([G_ref_f2sT(S(i)) G_ref_f2sT(mod(S(i),4)+1)]',S(i))=this(S(i)); else @@ -345,16 +329,13 @@ end end -%% Aktualisieren der VaterSohn Beziehung ! sollte nur ausgeführt werden wenn wirklich möglich +%% Aktualisieren der VaterSohn Beziehung ! sollte nur ausgefuehrt werden wenn wirklich moeglich function updateF2S(ele) %Vater Sohn Beziehungen richtig Setzen global G_ref_f2s; global G_ref_f2sT; global G_ref_t; global G_ref_tD; -% global G_ref_C; -% global G_ref_E; - if(G_ref_tD(ele)==1) %Wenn Element zum ersten Mal verfeinert wird G_ref_f2s(ele,:) = G_ref_f2sT; @@ -371,32 +352,24 @@ function updateF2S(ele) pos=find(G_ref_f2s(org,:)==ele,1); if(G_ref_t(ele)==3) -% G_ref_f2s(org,:) -% pos=find(G_ref_f2s(org,:)==ele,1) -% G_ref_f2sT - if(pos==1) G_ref_f2s(org,[1 4]) = G_ref_f2sT([1 3]); else G_ref_f2s(org,[2 3]) = G_ref_f2sT([3 1]); end -% G_ref_f2s(org,:) -% plotShape(G_ref_C,G_ref_E(G_ref_f2s(org,:),:),'t') else if(pos==1) G_ref_f2s(org,[1 2]) = G_ref_f2sT([1 2]); else G_ref_f2s(org,[3 4]) = G_ref_f2sT([2 1]); end -% G_ref_f2s(org,:) -% plotShape(G_ref_C,G_ref_E(G_ref_f2s(org,:),:),'t') end G_ref_t(G_ref_f2sT) = 0; end end -%% Berechnet die Flächeninhalte der Elemente +%% Berechnet die Flaecheninhalte der Elemente % function updateS(ele) % %Globale Variabelen aufbauen % global G_ref_E; diff --git a/src/test_sol.m b/src/test_sol.m index f9dee9d..41ab12a 100644 --- a/src/test_sol.m +++ b/src/test_sol.m @@ -9,7 +9,7 @@ mex -o mex_build_V.cpp slpRectangle.cpp CFLAGS="\$CFLAGS -fopenmp" CXXFLAGS="\$ % Test ausführen %Anzahl der Schritte oder wenn groeßer als 40 der Elemente -steps = 10^4; +steps = 4; %Art der Berechnungen type = [1 3 2]; zeta = { [2 2 2] [2 2 2] [2 2 2]}; -- 2.47.3

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