Finite Element Methods

We use the linear finite element method for solving the Poisson equation as an example to illustrate the main ingredients of finite element methods. We recommend to read

Contents

Variational formulation

The classic formulation of the Poisson equation reads as

$$ - \Delta u = f \hbox{ in } \Omega, \qquad u = g_D \hbox{ on } \Gamma _D, \qquad \nabla u\cdot n = g_N \hbox{ on } \Gamma _N, $$

where $\partial \Omega = \Gamma _D\cup \Gamma _N$ and $\Gamma _D\cap \Gamma _N=\emptyset$. We assume $\Gamma _D$ is closed and $\Gamma _N$ open.

Denoted by $H_{g_D}^1(\Omega)=\{v\in L^2(\Omega), \nabla v\in L^2(\Omega) \hbox{ and } v|_{\Gamma _D} = g_D\}$. Multiplying the Poisson equation by a test function $v$ and using integration by parts, we obtain the weak formulation of the Poisson equation: find $u\in H_{g_D}^1(\Omega)$ such that for all $v\in H_{0_D}^1$:

$$ a(u,v) := \int _{\Omega} \nabla u\cdot \nabla v\, {\rm dxdy} = \int _{\Omega} fv \, {\rm dxdy} + \int _{\Gamma _N} g_N v \,{dS}.$$

Let $\mathcal T$ be a triangulation of $\Omega$. We define the linear finite element space on $\mathcal T$ as

$$ \mathcal V_{\mathcal T} = \{v\in C(\bar \Omega) : v|_{\tau}\in \mathcal P_k, \forall \tau \in \mathcal T\}. $$

where $\mathcal P_k$ is the polynomial space with degree $\leq k$.

The finite element method for solving the Poisson equation is to find $u\in \mathcal V_{\mathcal T}\cap H_{g_D}^1(\Omega)$ such that for all $v\in \mathcal V_{\mathcal T}\cap H_{0_D}^1(\Omega)$:

$$ a(u,v) = \int _{\Omega} fv \, {\rm dxdy} + \int _{\Gamma _N} g_N v \,{dS}.$$

Finite element space

We take linear finite element spaces as an example. For each vertex $v_i$ of $\mathcal T$, let $\phi _i$ be the piecewise linear function such that $\phi _i(v_i)=1$ and $\phi _i(v_j)=0$ when $j\neq i$. The basis function in 1-D and 2-D is illustrated below. It is also called hat function named after the shape of its graph.

x = 0:1/5:1;
u = zeros(length(x),1);
u(2) = 1;
figure;
set(gcf,'Units','normal'); set(gcf,'Position',[0,0,0.5,0.3]);
subplot(1,2,1); hold on;
plot(x,0,'k.','MarkerSize',18);
plot(x,u,'-','linewidth',1.2);
subplot(1,2,2); hold on;
for k = 1:length(x)
    u = zeros(length(x),1); u(k) = 1;
    plot(x,0,'k.','MarkerSize',18);
    plot(x,u,'-','linewidth',1.2);
end

2-D hat basis

clf; set(gcf,'Units','normal'); set(gcf,'Position',[0,0,0.5,0.4]);
[node,elem] = squaremesh([0,1,0,1],0.25);
u = zeros(size(node,1),1);
u(12) = 1;
showmesh(node,elem,'facecolor','none'); hold on;
showsolution(node,elem,u,[30,26],'facecolor','g','facealpha',0.5,'edgecolor','k');

Then it is easy to see $\mathcal V_{\mathcal T}$ is spanned by $\{\phi _i\}_{i=1}^{N}$ and thus for a finite element function $v=\sum _{i=1}^Nv_i\phi _i$.

Progamming of Finite Element Methods


Published with MATLAB® R2015b