From bca146e82e8d3a160bfa48186f79eb49486815aa Mon Sep 17 00:00:00 2001 From: Peter Schaefer Date: Mon, 22 Apr 2013 16:16:33 +0200 Subject: [PATCH] =?utf8?q?[doc]=20Fehler=20weiter=20ausgebessert=20[src]?= =?utf8?q?=20Zeitmessung=20nur=20f=C3=BCr=20Aufstellen=20der=20Matrix?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- doc/doc.bib | 159 +++++++++++++++++++++++++------------------------- doc/doc.pdf | Bin 738557 -> 739810 bytes doc/doc.tex | 117 +++++++++++++++++++------------------ src/compute.m | 5 +- 4 files changed, 142 insertions(+), 139 deletions(-) diff --git a/doc/doc.bib b/doc/doc.bib index c8ee552..0ca6e71 100644 --- a/doc/doc.bib +++ b/doc/doc.bib @@ -1,95 +1,94 @@ +@comment{x-kbibtex-personnameformatting=<%l><, %f>} + % This file was created with JabRef 2.7b. % Encoding: UTF-8 -@ARTICLE{bor:errbox, - author = {Börm, Steffen and Grasedyck, Lars}, - title = {Low-Rank Approximation of Integral Operators by Interpolation}, - journal = {Computing}, - year = {2004}, - volume = {72}, - pages = {325-332}, - abstract = {A central component of the analysis of panel clustering techniques - for the approximation of integral operators is the so-called η -admissibility - condition “ min {diam(τ),diam(σ)} ≤ 2ηdist(τ,σ)” that ensures that - the kernel function is approximated only on those parts of the domain - that are far from the singularity. Typical techniques based on a - Taylor expansion of the kernel function require a subdomain to be - “far enough” from the singularity such that the parameter η has to - be smaller than a given constant depending on properties of the kernel - function. In this paper, we demonstrate that any η is sufficient - if interpolation instead of Taylor expansion␣is␣used for the kernel - approximation, which paves the way for grey-box panel clustering - algorithms.}, - doi = {10.1007/s00607-003-0036-0}, - issn = {0010-485X}, - issue = {3-4}, - owner = {treecity}, - publisher = {Springer-Verlag}, - timestamp = {2013.02.11}, - url = {http://dx.doi.org/10.1007/s00607-003-0036-0} +@article{bor:errbox, + abstract = "A central component of the analysis of panel clustering techniques for the approximation of integral operators is the so-called η -admissibility condition `` min {diam(τ),diam(σ)} ≤ 2ηdist(τ,σ)'' that ensures that the kernel function is approximated only on those parts of the domain that are far from the singularity. Typical techniques based on a Taylor expansion of the kernel function require a subdomain to be ``far enough'' from the singularity such that the parameter η has to be smaller than a given constant depending on properties of the kernel function. In this paper, we demonstrate that any η is sufficient if interpolation instead of Taylor expansion␣is␣used for the kernel approximation, which paves the way for grey-box panel clustering algorithms.", + author = "B{\"o}rm, Steffen and Grasedyck, Lars", + doi = "10.1007/s00607-003-0036-0", + issn = "0010-485X", + issue = "3-4", + journal = "Computing", + owner = "treecity", + pages = "325--332", + publisher = "Springer-Verlag", + timestamp = "2013.02.11", + title = "{Low-Rank Approximation of Integral Operators by Interpolation}", + url = "http://dx.doi.org/10.1007/s00607-003-0036-0", + volume = "72", + year = "2004" +} + +@article{dor:adapt, + abstract = "We construct a converging adaptive algorithm for linear elements applied to Poisson's equation in two space dimensions. Starting from a macro triangulation, we describe how to construct an initial triangulation from a priori information. Then we use a posteriori error estimators to get a sequence of refined triangulations and approximate solutions. It is proved that the error, measured in the energy norm, decreases at a constant rate in each step until a prescribed error bound is reached. Extensions to higher-order elements in two space dimensions and numerical results are included.", + author = "D{\"o}rfler, Willy", + copyright = "Copyright © 1996 Society for Industrial and Applied Mathematics", + issn = "00361429", + journal = "SIAM Journal on Numerical Analysis", + jstor_articletype = "research-article", + jstor_formatteddate = "Jun., 1996", + number = "3", + owner = "treecity", + pages = "pp. 1106--1124", + publisher = "Society for Industrial and Applied Mathematics", + timestamp = "2012.10.04", + title = "{A Convergent Adaptive Algorithm for Poisson's Equation}", + url = "http://www.jstor.org/stable/2158497", + volume = "33", + year = "1996" +} + +@article{fer:errbem, + author = "Ferraz-Leite, Samuel and Praetorius, Dirk", + journal = "Computing", + number = "4", + pages = "135--162", + title = "{Simple a posteriori error estimators fot the h-version of the boundary element method}", + volume = "83", + year = "2008" } -@ARTICLE{dor:adapt, - author = {Dörfler, Willy}, - title = {{A Convergent Adaptive Algorithm for Poisson's Equation}}, - journal = {SIAM Journal on Numerical Analysis}, - year = {1996}, - volume = {33}, - pages = {pp. 1106--1124}, - number = {3}, - abstract = {We construct a converging adaptive algorithm for linear elements applied - to Poisson's equation in two space dimensions. Starting from a macro - triangulation, we describe how to construct an initial triangulation - from a priori information. Then we use a posteriori error estimators - to get a sequence of refined triangulations and approximate solutions. - It is proved that the error, measured in the energy norm, decreases - at a constant rate in each step until a prescribed error bound is - reached. Extensions to higher-order elements in two space dimensions - and numerical results are included.}, - copyright = {Copyright © 1996 Society for Industrial and Applied Mathematics}, - issn = {00361429}, - jstor_articletype = {research-article}, - jstor_formatteddate = {Jun., 1996}, - owner = {treecity}, - publisher = {Society for Industrial and Applied Mathematics}, - timestamp = {2012.10.04}, - url = {http://www.jstor.org/stable/2158497} +@techreport{mai:3dbem, + author = "Maischak, Matthias", + institution = "Institut f{\"u}r Angewandte Mthematik, University of Hannover", + month = jul, + title = "{The analytical computation of the Galerkin elements for the Laplace, Lam{\'e} and Helmholtz equation in 3D-BEM}", + year = "2000" } -@ARTICLE{fer:errbem, - author = {Ferraz-Leite, Samuel and Praetorius, Dirk}, - title = {{Simple a posteriori error estimators fot the h-version of the boundary - element method}}, - journal = {Computing}, - year = {2008}, - volume = {83}, - pages = {135--162}, - number = {4} +@book{pla:nummat, + author = "Plato, Robert", + owner = "treecity", + publisher = "Vieweg Verlag", + timestamp = "2013.02.11", + title = "{Numerische Mathematik kompakt}", + year = "2006" } -@TECHREPORT{mai:3dbem, - author = {Maischak, Matthias}, - title = {{The analytical computation of the Galerkin elements for the Laplace, - Lam{\'e} and Helmholtz equation in 3D-BEM}}, - institution = {Institut f{\"u}r Angewandte Mthematik, University of Hannover}, - year = {2000}, - month = jul +@unpublished{pra:hmat, + author = "Praetorius, Dirk", + owner = "treecity", + timestamp = "2013.02.11", + title = "{Hierarchische Matrizen und Fast Multipole Method}", + year = "2009" } -@BOOK{pla:nummat, - title = {{Numerische Mathematik kompakt}}, - publisher = {Vieweg Verlag}, - year = {2006}, - author = {Plato, Robert}, - owner = {treecity}, - timestamp = {2013.02.11} +@phdthesis{fer:fehlsym, + author = "Ferraz-Leite, Samuel", + title = "{A posteriori Fehlersch{\"a}tzer f{\"u}r die Symmsche Integralgleichung}" } -@UNPUBLISHED{pra:hmat, - author = {Praetorius, Dirk}, - title = {Hierarchische Matrizen und Fast Multipole Method}, - year = {2009}, - owner = {treecity}, - timestamp = {2013.02.11} +@incollection{stb:fem, + author = "Steinbach, Olaf", + booktitle = "{Numerische N{\"a}herungsverfahren f{\"u}r elliptische Randwertprobleme}", + doi = "10.1007/978-3-322-80054-1_10", + isbn = "978-3-519-00436-3", + pages = "212--225", + publisher = "Vieweg+Teubner Verlag", + series = "{Advances in Numerical Mathematics}", + title = "{Randelemente}", + url = "http://dx.doi.org/10.1007/978-3-322-80054-1_10", + year = "2003" } diff --git a/doc/doc.pdf b/doc/doc.pdf index 1912e050b74725a26c0ffee70a0535e24cbbb82d..f710e68bb55b0422a0571ce9b7f549a73ff5756b 100644 GIT binary patch delta 453821 zcmV(kK=r@f7dIqrd38Vt?R?4l`ME=Z2hj0@NVNnec7~*8RwO)=S^#w zkXF~{@QiXy!9!g~yp%*Ksa*Qc@xNzJj%UwiX#&-o$=MiD2@QQ)*`JM*l`|fO7iV(R zw46!lx$^Y_$E~#Fu0D%J0b_gtQ&3zf1EZD92;pUY1tqR!@bw3Le_lypq5jWCK?Cb; 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zIQz@_4z7;G0l6@kpqdMkaTek-#y6VSkxV5lBo_~o_^|cXF8$xcevZB*8fK?G)?+3g zcIEpM4XgScJUjc$_ES!qd1koVZp~A^f64_jn{lF>ryM$K*4fPrQuCqBQ}#dO#97Um z=2J+7Q}*Ax`RLtP>hHUc&d}^zn7ViNySi^wN`IGSb1uJ6ao?%??(zQ*aZI?!9^qMM ax1V|Td1uZ%G1U2lR4SY>VV^_x4gVjn_Q@^) diff --git a/doc/doc.tex b/doc/doc.tex index 99d9e10..360a635 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -270,7 +270,7 @@ wobei $\varDelta u := \partial_x^2u+\partial_y^2u+\partial_z^2u$ den Laplace-Ope % wobei $V := \gamma_0\tilde V \in L(H^{-1/2}(\Gamma),H^{1/2}(\Gamma))$. Ziel ist es nun aus \todo{$(5)$} eine Funktion $\phi$ zu bestimmen, die die obige Gleichung erfüllt, denn dann ist $\tilde V\phi$ die Lösung des Problems. \subsection{Galerkin-Verfahren} -Die Fundamentallösung des Laplaceoperators \todo{cite Steinbach} ist für $\Omega \subset \R^3$ gegeben durch +Die Fundamentallösung des Laplaceoperators \todo{\cite[Theorem 12]{stb:fem}} ist für $\Omega \subset \R^3$ gegeben durch \begin{align*} % \label{math:slp:fundamental} G(\bs z) := \frac 1 {4 \pi \abs{\bs z}}. \end{align*} @@ -288,7 +288,7 @@ In der schwachen Formulierung lautet die Laplace-Gleichung - \varDelta u &= 0 \quad \text{ in } H^{-1}(\Omega), \\ \gamma_0 u &= g \quad \text{ auf }H^{1/2}(\Gamma). \end{align*} -Gemäß \todo{cite Steinbach} kann $\tilde V$ auch als Operator $\tilde V \in L(H^{-1/2}(\Gamma),H^{1}(\Omega))$ aufgefasst werden, und es gilt +Gemäß \todo{\cite[Kapitel 6.2 und 6.3]{stb:fem}} kann $\tilde V$ auch als Operator $\tilde V \in L(H^{-1/2}(\Gamma),H^{1}(\Omega))$ aufgefasst werden, und es gilt \begin{align*} -\varDelta\tilde V\phi = 0 \in H^{-1}(\Omega) \quad \text{für alle }\phi \in H^{-1/2}(\Gamma) \end{align*} @@ -550,13 +550,14 @@ Für eine Partition $\T_{\ell}$ bezeichnen wir mit $\widehat \T_{\ell}$ jene Par Für das Verständnis des folgenden Algorithmus benötigen wir noch einige Beobachtungen: \begin{itemize} \item Sind $e,\tilde e \in \E_T$ Kanten von $T \in \T_{\ell}$ mit $e \cap \tilde e = \emptyset$, so liegen diese gegenüber und haben insbesondere dieselbe Länge. Falls $e$ verfeinert werden soll, muss zwingend auch $\tilde e$ verfeinert werden. - \item Es bezeichnet $\S_{\ell} := \{ (e,T) ~|~ T \in \T_{\ell}, e \in \E_T \}$ die Menge aller Kanten von $\T_{\ell}$, wobei es aufgrund der maximal drei Knoten auf einer Kante vorkommen kann, dass eine Kante $e \in \S_{\ell}$, die im inneren von $\Gamma$ liegt, nur zu einem Element $T\in\T_{\ell}$ gehört und nicht zu zwei. In diesem Fall gibt es eine Kante $\tilde e \in \S_{\ell}$ in der $e$ ganz enthalten ist. Sollte $e$ verfeinert werden, so muss zwingend auch $\tilde e$ verfeinert werden, damit nicht mehr als drei Knoten auf der Kante $\tilde e$ entstehen. + \item Es bezeichnet $\S_{\ell} := \{ (e,T) ~|~ T \in \T_{\ell}, e \in \E_T \}$ die Menge aller Elemente mit zugehörigen Kanten. + Hierbei kann es aufgrund der maximal drei Knoten auf einer Kante vorkommen, dass eine Kante $e \in \E_{\ell}$, die im inneren von $\Gamma$ liegt, nur zu einem Element $T\in\T_{\ell}$ gehört und nicht zu zwei. In diesem Fall gibt es ein Element $\hat T\in \T_{\ell}$ mit Kante $\hat e \in \E_{\ell}$ in der $e$ ganz enthalten ist. Sollte $(e,T)$ verfeinert werden, so muss zwingend auch $(\hat e,\hat T)$ verfeinert werden, damit nicht mehr als drei Knoten auf der Kante $\hat e$ entstehen. \end{itemize} \begin{alg}[Verfeinern] \label{alg:refine} Sei $\T_{\ell}$ eine Partition und $\sqcap_{\ell} \subseteq \S_{\ell}$ eine Menge markierter Kanten. Nun sei $\sqcap_{\ell}^{(0)}:=\sqcap_{\ell}$ und $i=0$. Dann gehe so vor: \begin{enumerate} \renewcommand{\theenumi}{(\roman{enumi})} \item \label{alg:refine:first} - $\sqcap_{\ell}^{(i+1/2)} := \sqcap_{\ell}^{(i)} \cup \{ (e,T) \in \S_{\ell}\backslash\sqcap_{\ell}^{(i)} ~|~ \exists (\tilde e,\tilde T) \in \sqcap_{\ell}^{(i)}$ mit $e \supsetneqq \tilde e \}$ + $\sqcap_{\ell}^{(i+1/2)} := \sqcap_{\ell}^{(i)} \cup \{ (e,T) \in \S_{\ell}\backslash\sqcap_{\ell}^{(i)} ~|~ \exists (\hat e,\hat T) \in \sqcap_{\ell}^{(i)}$ mit $e \supsetneqq \hat e \}$ \item $\sqcap_{\ell}^{(i+1)} := \sqcap_{\ell}^{(i+1/2)} \cup \{(e,T) \in \S_{\ell}\backslash\sqcap_{\ell}^{(i+1/2)} ~|~ \exists (\tilde e,\tilde T) \in \sqcap_{\ell}^{(i+1/2)}$ mit $T=\tilde T$ und $e\cap \tilde e=\emptyset\}$ \item @@ -663,11 +664,11 @@ Unter einer Quadratur verstehen wir die approximative Berechnung eines Integrals \end{align*} durch die Summe \begin{align*} - \Q(f) &:= \sum_{k=0}^n w_k f(t_k) . + \Q_n(f) &:= \sum_{k=0}^n w_k f(t_k) . \end{align*} Die $t_k$ sind hierbei die Knoten, die $w_k$ Gewichte. -Eine Quadratur heißt exakt für eine Funktion $f$, falls $\Q(f) = \int_0^1 f(x) dx$ ist.\\ -Eine Quadratur hat den Exaktheitsgrad $m$, wenn sie für alle Polynome bis zum Grad $m$ exakt ist.\\ +Eine Quadratur heißt exakt für eine Funktion $f$, falls $\Q_n(f) = \int_0^1 f(x) dx$ ist.\\ +Weiterhin hat eine Quadratur den Exaktheitsgrad $m$, wenn sie für alle Polynome bis zum Grad $m$ exakt ist.\\ Eine Quadratur heißt interpolatorisch (vom Grad $n$), falls für jede integrierbare, auf $(0,1)$ stetige Funktion $f$ \begin{align*} \Q_n(f) &:= \sum_{k=0}^n w_k f(t_k) = \sum_{k=0}^n w_k p(t_k) = \int_0^1 p(x) dx @@ -1182,7 +1183,7 @@ Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \end{align*} für festes $\bs x \in \R^3$, inklusive einer geeigneten Parametrisierung $\gamma_k : [0,1]^2 \to T_k$ \begin{align*} - \tilde F_{\bs x}(\bs s) = \int_0^{s_1} \int_0^{s_2} \frac 1 {\abs{\bs x -\gamma_k(\bs \lambda)}} d\lambda_2 d\lambda_1. + \tilde F_{\bs x} = \int \int \frac 1 {\abs{\bs x -\gamma_k(\bs \lambda)}} d\lambda_2 d\lambda_1. \end{align*} Die Stammfunktion ist auch mit der Parametrisierung stetig. Daher gilt für $T_k$ mit geringem Durchmesser $\diam(T_k)$ aber $\tilde F_{\bs x}(\lambda) \approx \tilde F_{\bs x}(\tilde \lambda)$ für alle $\lambda,\tilde \lambda \in [0,1]^2$, wodurch für den Ausdruck \begin{align*} @@ -1552,7 +1553,8 @@ 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} -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. +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} @@ -1685,94 +1687,94 @@ Dann gilt nach \cite{mai:3dbem} \subsection{Bestimmtes Integral} -Mithilfe der Stammfunktionen $F_p$ und $F_o$ können wir nun das Integral aus \eqref{math:analy:int} für orthogonal und parrallel liegenden achsenorientierte Rechtecke $T$ und $\tilde T$ berechnen. Im Folgenden sei $F_{p/o} \in \{F_p,F_o\}$. +Mithilfe der Stammfunktionen $F_p$ und $F_o$ können wir nun das Integral aus \eqref{math:analy:int} für orthogonal und parallel liegenden achsenorientierte Rechtecke $T$ und $\tilde T$ analytisch exakt lösen. Die Lösung ist für $F_{p/o} \in \{F_p,F_o\}$, geeignete Grenzen $s_1,s_2,\tilde s_1,\tilde s_2\in\R^+$ und $\bs \delta \in \R^3$ gegeben durch \begin{align*} - \int_T& \int_{\tilde T} \frac{1}{|x-y|} ds_y ds_x\\ -&= \int_0^{k_1}\int_0^{k_2}\int_0^{\tilde k_1}\int_0^{\tilde k_2} + \int_{T_j}& \int_{T_k} \frac{1}{|\bs x-\bs y|} ds_{\bs y} ds_{\bs x}\\ +&= \int_0^{s_1}\int_0^{s_2}\int_0^{\tilde s_1}\int_0^{\tilde s_2} \dif{}{y_2} \dif{}{y_1} \dif{}{x_2} \dif{}{x_1} - F_{p/o}(x_1,x_2,y_1,y_2) + F_{p/o}(x_1,x_2,y_1,y_2,\bs \delta) dy_2 dy_1 dx_2 dx_1\\ % -=& \int_0^{k_1}\int_0^{k_2}\int_0^{\tilde k_1} \big( +=& \int_0^{s_1}\int_0^{s_2}\int_0^{\tilde s_1} \big( \dif{}{y_1} \dif{}{x_2} \dif{}{x_1} - F_{p/o}(x_1,x_2,y_1,\tilde k_2) \\ + F_{p/o}(x_1,x_2,y_1,\tilde s_2,\bs \delta) \\ &- \dif{}{y_1} \dif{}{x_2} \dif{}{x_1} - F_{p/o}(x_1,x_2,y_1,0) \big) + F_{p/o}(x_1,x_2,y_1,0,\bs \delta) \big) dy_1 dx_2 dx_1\\ % -=& \int_0^{k_1}\int_0^{k_2}\big( +=& \int_0^{s_1}\int_0^{s_2}\big( \dif{}{x_2} \dif{}{x_1} - F_{p/o}(x_1,x_2,\tilde k_1,\tilde k_2) + F_{p/o}(x_1,x_2,\tilde s_1,\tilde s_2,\bs \delta) - \dif{}{x_2} \dif{}{x_1} - F_{p/o}(x_1,x_2,\tilde k_1,0) \\ + F_{p/o}(x_1,x_2,\tilde s_1,0,\bs \delta) \\ &- \dif{}{x_2} \dif{}{x_1} - F_{p/o}(x_1,x_2,0,\tilde k_2) + F_{p/o}(x_1,x_2,0,\tilde s_2,\bs \delta) + \dif{}{x_2} \dif{}{x_1} - F_{p/o}(x_1,x_2,0,0)\big) + F_{p/o}(x_1,x_2,0,0,\bs \delta)\big) dx_2 dx_1\\ % -=& \int_0^{k_1}\big( +=& \int_0^{s_1}\big( \dif{}{x_1} - F_{p/o}(x_1,k_2,\tilde k_1,\tilde k_2) + F_{p/o}(x_1,s_2,\tilde s_1,\tilde s_2,\bs \delta) - \dif{}{x_1} - F_{p/o}(x_1,k_2,\tilde k_1,0) \\ + F_{p/o}(x_1,s_2,\tilde s_1,0,\bs \delta) \\ &- \dif{}{x_1} - F_{p/o}(x_1,k_2,0,\tilde k_2) + F_{p/o}(x_1,s_2,0,\tilde s_2,\bs \delta) + \dif{}{x_1} - F_{p/o}(x_1,k_2,0,0)\\ + F_{p/o}(x_1,s_2,0,0,\bs \delta)\\ &- %% \dif{}{x_1} - F_{p/o}(x_1,0,\tilde k_1,\tilde k_2) + F_{p/o}(x_1,0,\tilde s_1,\tilde s_2,\bs \delta) + \dif{}{x_1} - F_{p/o}(x_1,0,\tilde k_1,0) \\ + F_{p/o}(x_1,0,\tilde s_1,0,\bs \delta) \\ &+ \dif{}{x_1} - F_{p/o}(x_1,0,0,\tilde k_2) + F_{p/o}(x_1,0,0,\tilde s_2,\bs \delta) - \dif{}{x_1} - F_{p/o}(x_1,0,0,0)\big) + F_{p/o}(x_1,0,0,0,\bs \delta)\big) dx_1\\ % =& - F_{p/o}(k_1,k_2,\tilde k_1,\tilde k_2) + F_{p/o}(s_1,s_2,\tilde s_1,\tilde s_2,\bs \delta) - - F_{p/o}(k_1,k_2,\tilde k_1,0) + F_{p/o}(s_1,s_2,\tilde s_1,0,\bs \delta) - - F_{p/o}(k_1,k_2,0,\tilde k_2) + F_{p/o}(s_1,s_2,0,\tilde s_2,\bs \delta) + - F_{p/o}(k_1,k_2,0,0)\\ + F_{p/o}(s_1,s_2,0,0,\bs \delta)\\ &- - F_{p/o}(k_1,0,\tilde k_1,\tilde k_2) + F_{p/o}(s_1,0,\tilde s_1,\tilde s_2,\bs \delta) + - F_{p/o}(k_1,0,\tilde k_1,0) + F_{p/o}(s_1,0,\tilde s_1,0,\bs \delta) + - F_{p/o}(k_1,0,0,\tilde k_2) + F_{p/o}(s_1,0,0,\tilde s_2,\bs \delta) - - F_{p/o}(k_1,0,0,0)\\ + F_{p/o}(s_1,0,0,0,\bs \delta)\\ &- %% - F_{p/o}(0,k_2,\tilde k_1,\tilde k_2) + F_{p/o}(0,s_2,\tilde s_1,\tilde s_2,\bs \delta) - - F_{p/o}(0,k_2,\tilde k_1,0) + F_{p/o}(0,s_2,\tilde s_1,0,\bs \delta) - - F_{p/o}(0,k_2,0,\tilde k_2) + F_{p/o}(0,s_2,0,\tilde s_2,\bs \delta) + - F_{p/o}(0,k_2,0,0)\\ + F_{p/o}(0,s_2,0,0,\bs \delta)\\ &- - F_{p/o}(0,0,\tilde k_1,\tilde k_2) + F_{p/o}(0,0,\tilde s_1,\tilde s_2,\bs \delta) + - F_{p/o}(0,0,\tilde k_1,0) + F_{p/o}(0,0,\tilde s_1,0,\bs \delta) + - F_{p/o}(0,0,0,\tilde k_2) + F_{p/o}(0,0,0,\tilde s_2,\bs \delta) - - F_{p/o}(0,0,0,0) + F_{p/o}(0,0,0,0,\bs \delta) \end{align*} @@ -2117,7 +2119,7 @@ Im Folgenden werden wir die Laplace-Gleichung \begin{align}\label{math:bsp:Quad:gls} \begin{aligned} - \varDelta u &= 0 \quad\text{ in } \Omega \subset \R^3, \\ -u &= g \quad \text{ auf }\Gamma, +u_{|\Gamma} &= 1 \quad \text{ auf }\Gamma, \end{aligned} \end{align} mit dem Startnetz aus Abbildung \ref{fig:mesh:2DQuad:start}, einem Quadrat in der $z=0$ Ebene genauer betrachten. Die vier Eckpunkte des Quadrats sind hierbei gegeben durch @@ -2136,7 +2138,7 @@ Die Lösung $\phi$ kann nicht exakt bestimmt werden, weist aber an den vier Eckp \subsubsection{Vergleich verschiedener Verfeinerungsstrategien} \noindent -Zunächst wollen wir drei Verfeinerungs-Strategien genauer untersuchen. Hierzu betrachten wir zum einen die Strategie "`uniform"' ($\theta=1,\nu=0$) bei der das verfeinerte Netz $\T_{\ell+1}$ durch isotrope Verfeinerung aller Elemente entsteht, also jedes Element wird in vier gleich Große Elemente geteilt. In der zweiten Strategie "`isotrop"' ($\theta=0.5,\nu=0$) werden wir zulassen, dass nicht alle Elemente verfeinert werden, also nur eine Teilmenge wird jeweils in vier gleich große Elemente geteilt. Und in der letzten Strategie "`anisotrop"' ($\theta=0.5,\nu=0.5$) werden wir außerdem anisotrope Teilungen zulassen, also ein Teil der Elemente wird geeignet in zwei oder vier gleich Große Elemente geteilt. Für alle drei Strategien werden wir den Algorithmus \ref{alg:adapt} mit entsprechenden Parametern $\theta,\nu$ verwenden. +Zunächst wollen wir drei Verfeinerungs-Strategien genauer untersuchen. Hierzu betrachten wir zum einen die Strategie "`uniform"' ($\theta=1,\nu=0$) bei der das verfeinerte Netz $\T_{\ell+1}$ durch isotrope Verfeinerung aller Elemente entsteht, also jedes Element wird in vier gleich Große Elemente geteilt. In der zweiten Strategie "`adaptiv isotrop"' ($\theta=0.5,\nu=0$) werden wir zulassen, dass nicht alle Elemente verfeinert werden, also nur eine Teilmenge wird jeweils in vier gleich große Elemente geteilt. Und in der letzten Strategie "`adaptiv anisotrop"' ($\theta=0.5,\nu=0.5$) werden wir außerdem anisotrope Teilungen zulassen, also ein Teil der Elemente wird geeignet in zwei oder vier gleich Große Elemente geteilt. Für alle drei Strategien werden wir den Algorithmus \ref{alg:adapt} mit entsprechenden Parametern $\theta,\nu$ zum Lösen von \eqref{math:bsp:Quad:gls} verwenden. \begin{figure}[ht] @@ -2180,40 +2182,40 @@ Zunächst wollen wir drei Verfeinerungs-Strategien genauer untersuchen. Hierzu b In Abbildung \ref{fig:2DQuad:verfeinern:err} betrachten wir für die jeweilige Strategie jeweils die Fehlerschätzer $\tilde \mu$ und $\eta$, sowie den Fehler gegenüber der tatsächlichen Lösung. Die Werte wurden jeweils über die Elementanzahl des $\T_{\ell}$ Netzes aufgetragen. Anhand der \figLineA[en] Linien erkennen wir, dass der Fehler bei der "`uniformen"' Strategie mit etwa $\O(N^{-1/4})$ gegen $0$ strebt. Gerechnet wurde hier bis zu einer Elementanzahl des $\T_{\ell}$ Netzes von etwa 3000, für das der Fehler der Energienorm im Bereich von 1 liegt. \noindent -Anhand der Linien in \figLineB[], beobachten wir eine schnellere Konvergenz der "`isotropen"' Strategie. Denn hier lässt sich eine Konvergenzrate von $\O(N^{-1/2})$ gegen 0 erkennen. Bei dieser Strategie erreichen wir für die gleiche Elementanzahl des $\T_{\ell}$ Gitters schon einen Fehler im Bereich von 0.1. +Anhand der Linien in \figLineB[], beobachten wir eine schnellere Konvergenz der "`adaptiv isotropen"' Strategie. Denn hier lässt sich eine Konvergenzrate von $\O(N^{-1/2})$ gegen 0 erkennen. Bei dieser Strategie erreichen wir für die gleiche Elementanzahl des $\T_{\ell}$ Gitters schon einen Fehler im Bereich von 0.1. \noindent -Betrachten wir nun die Strategie "`anisotrop"' in \figLineC[], so beobachten wir eine kurzzeitig sehr starke Konvergenz, welche dann von unten gegen eine sehr gute Konvergenzrate von $\O(N^{-3/4})$ strebt. Hierbei erkennen wir, dass der Fehler der Energienorm schon im Bereich von 0.01 für die gleiche Elementanzahl liegt. Jedoch sehen wir auch, dass der Fehlerschätzer $\tilde \mu$ ab dieser Elementanzahl seine Konvergenzrate verliert und damit unzuverlässig wird. +Betrachten wir nun die Strategie "`adaptiv anisotrop"' in \figLineC[], so beobachten wir eine kurzzeitig sehr starke Konvergenz, welche dann von unten gegen eine sehr gute Konvergenzrate von $\O(N^{-3/4})$ strebt. Hierbei erkennen wir, dass der Fehler der Energienorm schon im Bereich von 0.01 für die gleiche Elementanzahl liegt. Jedoch sehen wir auch, dass der Fehlerschätzer $\tilde \mu$ ab dieser Elementanzahl seine Konvergenzrate verliert und damit unzuverlässig wird. \noindent Weiterhin können wir für alle drei Strategien anhand der Parallelität der Fehlerschätzer zum tatsächlichen Fehler, die Effektivität und Zuverlässigkeit der Fehlerschätzer erkennen. Sowie auch die Äquivalents des $h-h/2$ Schätzers zum lokalen $\tilde \mu$ Schätzers aufgrund der Parallelität zu beobachten ist. Außerdem beschreiben die Fehlerschätzer den tatsächlichen Fehler auch in der Größenordnung sehr gut. \noindent -In Abbildung \ref{fig:2DQuad:verfeinern:hminmax} betrachten wir bestimmte Eigenschaften zwischen den Seitenverhältnissen der Elemente aus dem $\T_{\ell}$ Netz für die jeweilige Strategie. $h_{\min}$ steht hierbei für die kürzere Seite eines Rechtecks $T \in \T_{\ell}$ und $h_{\max}$ für die längere der beiden Seiten. Wir werden nun das Verhältnis der kleinsten kurzen Seite gegenüber der größten langen Seite $\min(h_{\min}) / \max (h_{\max})$, das Verhältnis der kleinsten langen gegenüber der größten langen Seite $\min(h_{\max}) / \max (h_{\max})$, sowie das kleinste Verhältnis der kurzen gegenüber der langen Seiten $\min(h_{\min} /h_{\max})$ für die drei Strategien genauer betrachten. +In Abbildung \ref{fig:2DQuad:verfeinern:hminmax} betrachten wir bestimmte Eigenschaften zwischen den Seitenverhältnissen der Elemente aus dem $\T_{\ell}$ Netz für die jeweilige Strategie. $h_{\min}$ steht hierbei für die kürzere Seite eines Rechtecks $T \in \T_{\ell}$ und $h_{\max}$ für die längere der beiden Seiten. Weiterhin verstehen wir das Minimum $\min()$ und Maximum $\max()$ über alle Elemente $T\in\T_{\ell}$. Wir werden nun das Verhältnis der kleinsten kurzen Seite gegenüber der größten langen Seite $\min_{}(h_{\min}) / \max_{}(h_{\max})$, das Verhältnis der kleinsten langen gegenüber der größten langen Seite $\min_{}(h_{\max}) / \max_{}(h_{\max})$, sowie das kleinste Verhältnis der kurzen gegenüber der langen Seiten $\min_{}(h_{\min} /h_{\max})$ für die drei Strategien genauer betrachten. \noindent Bei der "`uniformen"' Strategie, dargestellt durch die \figLineA[e] Linien, sind wie erwartet alle Verhältnisse gleich 1 da alle Elemente deckungsgleich sind. \noindent -Bei der "`isotropen"' Strategie in \figLineB[] ist am Verhältnis $\min(h_{\min} /h_{\max}) = 1$ gut zu erkennen, dass alle Elemente Quadrate sind. Die beiden anderen Verhältnisse zeigen, dass für zunehmende Elementanzahl die Differenz der Elementgrößen zunimmt. +Bei der "`adaptiv isotropen"' Strategie in \figLineB[] ist am Verhältnis $\min_{}(h_{\min} /h_{\max}) = 1$ gut zu erkennen, dass alle Elemente Quadrate sind. Die beiden anderen Verhältnisse zeigen, dass für zunehmende Elementanzahl die Differenz der Elementgrößen zunimmt. \noindent -Anhand der \figLineC[] farbenen Linien, also der "`anisotropen"' Strategie beobachten wir, am kleiner werdenden Verhältnis $\min(h_{\min} /h_{\max})$, dass mit zunehmender Elementanzahl lange schmale Elemente entstehen. Am kleiner werden der anderen beiden Verhältnisse erkennen wir auch hier, dass die Differenz der Elementgrößen zunimmt. +Anhand der \figLineC[] farbenen Linien, also der "`adaptiv anisotropen"' Strategie beobachten wir, am kleiner werdenden Verhältnis $\min_{}(h_{\min} /h_{\max})$, dass mit zunehmender Elementanzahl lange schmale Elemente entstehen. Am kleiner werden der anderen beiden Verhältnisse erkennen wir auch hier, dass die Differenz der Elementgrößen zunimmt. \noindent Um auch die Stabilität der drei Strategien untersuchen zu können, sehen wir in der Abbildung \ref{fig:2DQuad:verfeinern:cond} die Konditionszahlen der $V_{\ell}$ Matrix in Abhängigkeit der Elementanzahl. \noindent -Weiterhin können wir in Abbildung \ref{fig:2DQuad:verfeinern:time} die benötigte Zeit pro Berechnungsschritt ablesen. Hierbei fällt auf, dass die Wahl der Strategie keinen Einfluss auf die benötigte Zeit hat, sondern nur die Anzahl der Elemente. Für die Berechnung mit 3000 Elementen benötigen alle drei Strategien etwa $10^4$ Sekunden, was fast drei Stunden entspricht. +Weiterhin können wir in Abbildung \ref{fig:2DQuad:verfeinern:time} die benötigte Zeit pro Berechnungsschritt ablesen. Zu einem Berechnungsschritt gehört das Aufstellen der Matrix $\hat{\bs V_{\ell}}$, berechnen der Kondition von $\hat{\bs V_{\ell}}$, berechnen der Galerkin-Lösung und berechnen der Fehlerschätzer. Hierbei fällt auf, dass die Wahl der Strategie keinen Einfluss auf die benötigte Zeit hat, sondern nur die Anzahl der Elemente. Für die Berechnung mit 3000 Elementen benötigen alle drei Strategien etwa $10^4$ Sekunden, was fast drei Stunden entspricht. \noindent -Diese Ergebnisse Zeigen also, dass die "`anisotrope"' Strategie die beste Konvergenzrate aufweist, wir dafür jedoch eine schlechtere Konditionszahl der Matrix in kauf nehmen müssen. Dies ist letztendlich auf die Größe und Forme der Elemente zurückzuführen. An dieser Stelle wollen wir noch zusätzlich Abbildung \ref{fig:mesh:2DQuad:steps} betrachten, welche das "`anisotrop"' verfeinerte Netz nach 12 Schritten darstellt. Denn hier erkennen wir sehr gut, dass diese Strategie das Netz insbesondere an den Singularitäten verfeinert. +Diese Ergebnisse Zeigen also, dass die "`adaptiv anisotrope"' Strategie die beste Konvergenzrate aufweist, wir dafür jedoch eine schlechtere Konditionszahl der Matrix in kauf nehmen müssen. Dies ist letztendlich auf die Größe und Forme der Elemente zurückzuführen. An dieser Stelle wollen wir noch zusätzlich Abbildung \ref{fig:mesh:2DQuad:steps} betrachten, welche das "`adaptiv anisotrop"' verfeinerte Netz nach 12 Schritten darstellt. Denn hier erkennen wir sehr gut, dass diese Strategie das Netz insbesondere an den Singularitäten verfeinert. \noindent Ziel wird es nun sein die Instabilitäten, die bei der analytischen Berechnung auftreten durch Quadratur zu vermeiden. Dafür werden wir vorher noch Berechnungen mit verschiedenen Quadraturgraden genauer untersuchen. \subsubsection{Vergleich verschiedener Quadraturgrade} -Bei der folgenden Berechnung werden wir alle Integrale von $\zeta_Q$-zulässigen Elemente durch die vorgestellte Gauss-Quadratur approximieren. Hierbei werden wir jeweils 1, 2, 4 oder 8 Auswertungsstellen verwenden. Alle vier Berechnungsarten werden auf den selben Netzen ausgeführt um die Ergebnisse nicht durch die Wahl des Netzes zu beeinflusse. Zum Berechnen des $\tilde \mu_{\ell}$-Schätzers, welcher die Verfeinerung steuert, werden wir in jedem Schritt die Lösung der Quadratur mit 8 Auswertungsstellen verwenden. +Bei der folgenden Berechnung werden wir wieder den Algorithmus \ref{alg:adapt} mit Parametern $\theta=0.5,\nu=0.5$ zum Lösen von \eqref{math:bsp:Quad:gls} verwenden, wobei alle Integrale von $\zeta_Q$-zulässigen Elemente durch die vorgestellte Gauss-Quadratur approximiert werden. Hierbei werden wir jeweils 1, 2, 4 oder 8 Auswertungsstellen verwenden. Alle vier Berechnungsarten werden auf den selben Netzen ausgeführt um die Ergebnisse nicht durch die Wahl des Netzes zu beeinflusse. Zum Berechnen des $\tilde \mu_{\ell}$-Schätzers, welcher die Verfeinerung steuert, werden wir in jedem Schritt die Lösung der Quadratur mit 8 Auswertungsstellen verwenden. \begin{figure}[ht] @@ -2419,7 +2421,8 @@ for(int i=0;i' num2str(i)]) %Matrix aufbauen -> MEX V_fine = mex_build_V(coo_fine,ele_fine,zeta{i},typ(i)); - + build_time = toc; %Testet auf Fehlerhafte Eintraege (NaN +/-Inf) [r c] = find(isnan(V_fine)~=isinf(V_fine)); if(~isempty(r)) @@ -169,7 +169,8 @@ for j = 1:times % eta = xd_fine'*A_fine*xd_fine; eta = abs(xe_fine-xe); - end_time = toc - start_time; +% end_time = toc - start_time; + end_time = build_time - start_time; dataS = [dataS ... typ(i) ... berechnet mit Typ -- 2.47.3

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