From 50d276043886d428b804516376de59f6b9528982 Mon Sep 17 00:00:00 2001 From: bangerth Date: Fri, 10 Dec 2010 04:04:50 +0000 Subject: [PATCH] Add Luca's text. git-svn-id: https://svn.dealii.org/trunk@22955 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-38/doc/intro.dox | 108 +++++++++++++++++++++++++ 1 file changed, 108 insertions(+) diff --git a/deal.II/examples/step-38/doc/intro.dox b/deal.II/examples/step-38/doc/intro.dox index eae18a9fea..2ecd427c90 100644 --- a/deal.II/examples/step-38/doc/intro.dox +++ b/deal.II/examples/step-38/doc/intro.dox @@ -7,6 +7,114 @@

Introduction

+In this example, we show how to solve a partial differential equation (PDE) +on a codimension one surface $\Gamma \subset \mathbb R^3$ +made of quadrilaterals, i.e. on a surface in 3d or a line in 2d. +We focus on the following elliptic second order PDE +@f{align*} +-\Delta_\Gamma u &= f \qquad \text{on } \qquad \Gamma,\\ +u &= g \qquad \text{on} \qquad \partial \Gamma, +@f} +which generalized the Laplace equation we have previously solved in several of +the early tutorial programs. Our implementation is based on step-4. step-34 +also solves problems on lower dimensional surfaces; however, there we only +consider integral equations that do not involve derivatives on the solution +variable, while here we actually have to investigate what it means to take +derivatives of a function only defined on a (possibly curved) surface. + +In order to define the above operator, we start by introducing some notations. +Let $\mathbf x_S:\hat S \rightarrow S$ be a parametrization of +a surface $S$ from a reference element $\hat S \subset \mathbb R^2$, +i.e. each point $\hat{\mathbf x}\in\hat S$ induces a point ${\mathbf + x}_S(\hat{\mathbf x}) \in S$. Then let +@f[ +G_S:= (D \mathbf{x}_S)^T \ D \mathbf{x}_S +@f] +denote the corresponding first fundamental form, where $D +\mathbf{x}_S=\left(\frac{\partial x_{S,i}(\hat{\mathbf x})}{\partial \hat x_j}\right)_{ij}$ is the +derivative (Jacobian) of the mapping. +In the following, $S$ will be either the entire surface $\Gamma$ or, +more convenient for the finite element method, any face $S \in +{\mathbb T}$, where ${\mathbb T}$ is a partition (triangulation) of $\Gamma$ +constituted of quadrilaterals. +We are now in position to define the tangential gradient of a function $v : S \rightarrow \mathbb +R$ by +@f[ +(\nabla_S v)\circ \mathbf x_S := D \mathbf x_S \ G_S^{-1} \ D (v \circ \mathbf x_S). +@f] +The surface Laplacian (also called the Laplace-Beltrami operator) is then +defined as $\Delta_S:= \nabla_S \cdot \nabla_S$. + +As usual, we are only interested in weak solutions for which we can use $C^0$ +finite elements (rather than requiring $C^1$ continuity as for strong +solutions). We therefore resort to the weak formulation +@f[ +\int_\Gamma \nabla_\Gamma u \cdot +\nabla_\Gamma v = \int_\Gamma f \ v \qquad \forall v \in H^1_0(\Gamma) +@f] +and take advantage of the partition ${\mathbb T}$ to further write +@f[ +\sum_{K\in {\mathbb T}}\int_K \nabla_K u \cdot \nabla_K v = \sum_{K\in + {\mathbb T}} \int_K f \ v \qquad \forall v \in H^1_0(\Gamma). +@f] +Moreover, each integral in the above expression is computed in the reference +element $\hat K:= [0,1]^2$ +so that +@f{align*} +&\int_{K} \nabla_{K} u \cdot \nabla_{K} v \\ +&= +\int_{\hat K} D(u \circ \mathbf x_K)^T G_K^{-1} (D \mathbf + x_K)^T D \mathbf x_K G_K^{-1} D(v \circ \mathbf x_K) \sqrt{\det + (G_K)} +\\ +&= +\int_{\hat K} D(u \circ \mathbf x_K)^T G_K^{-1} D(v \circ \mathbf x_K) \sqrt{\det + (G_K)} +@f} +and +@f[ +\int_{K} f \ v = \int_{\hat K} (f \circ \mathbf x_K) (v \circ \mathbf +x_K) \sqrt{\det + (G_K)}. +@f] +Finally, we use a quadrature formula defined by points $\{p_l\}_{l=1}^N\subset +\hat K$ and weights $\{w_l\}_{l=1}^N \subset \mathbb R^+_*$ to +evaluate the above integrals and +obtain +@f[\int_{K} \nabla_{K} u \cdot \nabla_{K} v \approx \sum_{l=1}^N + (D(u \circ \mathbf x_K)(p_l))^T G^{-1}(p_l) D(v \circ \mathbf x_K) +(p_l) \sqrt{\det (G(p_l))} \ w_l +@f] +and +@f[ +\int_{K} f \ v \approx \sum_{l=1}^N (f \circ \mathbf x_K)(p_l) \ (v \circ \mathbf x_K)(p_l) \sqrt{\det (G(p_l))} \ w_l. +@f] + + +Fortunately, deal.II has already all the tools to compute the above +expressions. +In fact, they barely differ from the ways in which we solve the usual +Laplacian, only requiring the surface coordinate mapping to be provided in the +constructor of the FEValues class. +This surface description given, in the codimension one surface case, the two +routines FEValues::shape_grad and FEValues::JxW +return +@f{align*} +\text{FEValues::shape\_grad}(i,l)&=D \mathbf x_K(p_l) G^{-1}(p_l)D(\varphi_i \circ \mathbf x_K) + (p_l) +\\ +\text{FEValues::JxW}(l) &= \sqrt{\det (G(p_l))} \ w_l. +@f} +This provides exactly the terms we need for our computations. + +On a more general note, details for the finite element approximation on +surfaces can be found for instance in +[Dziuk, in Partial differential equations and calculus of +variations 1357, Lecture Notes in Math., 1988] +and +[Bonito, Nochetto, and Pauletti, SIAM J. Numer. Anal. 48(5), 2010]. + +

The numerical approximation

Implementation

-- 2.39.5