Partial differential equations and the finite element method 3-FEA1.md

This part is devoted to introduce the Galerkin method and its important special case, the Finite element method.

Consider the general framework

Let V$V$be a Hilbert space, a(.,.):V×V→R$a(.,.):V\times V\to R$ a bilinear form and l∈V′$l\in V^{\prime }$. It is our task to find u∈V$u\in V$ such that
(1)

We assume that the bilinear form a(.,.)$a(.,.)$ is bounded and V-elliptic, i.e., that there exist constants Cb,Cel>0$C_b,C_{el}>0$ such that

and

Recall that the weak problem (1) has a unique solution by the Lax-Milgram lemma.

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The Galerkin method

As V$V$ is infinite dim function space. The Galerkin method is based on a sequence of finite dimensional subspaces {Vn}∞n=1⊂V$\{V_n{\}}_{n=1}^{\mathrm{\infty }}\subset V$, Vn⊂Vn+1$V_n\subset V_{n+1}$, that fill the space V$V$ in the limit. In each finite-dimensional space Vn$V_n$, problem (1) is solved exactly. It can be shown that under suitable assumptions the sequence of the approximate solutions {un}∞n=1$\{u_n{\}}_{n=1}^{\mathrm{\infty }}$ converges to the exact solution of problem (1).

Discrete problem

Find un∈Vn$u_n\in V_n$ such that
(2)

Lemma 1 (Unique solvability) Problem (2) has a unique solution un∈Vn$u_n\in V_n$,.

Proof: Recall Lax-Milgram lemma : Let V$V$ be a Hilbert space, a(.,.):V×V→R$a(.,.):V\times V\to R$ a bounded V-elliptic bilinear form and l∈V′$l\in V^{\prime }$ Then there exists a unique solution to the problem

.

Suppose space V$V$ has a finite basis {vn}Nnn=1$\{v_n{\}}_{n=1}^{N_n}$, then

where yj$y_j$s are unknown coefficients.

Denote Sn={a(vj,vi)}Ni,j=1$S_n=\{a(v_j,v_i){\}}_{i,j=1}^N$, Fn={l(vi)}$F_n=\{l(v_i)\}$, Yn={yj}$Y_n=\{y_j\}$, then
(3).

Lemma 2 (Positive definiteness of Sn$S_n$) Let Vn$V_n$ be a Hilbert space and a(.,.):V×V→R$a(.,.):V\times V\to R$a bilinear V-elliptic form. Then the stiffness matrix Sn$S_n$ , of the discrete problem (3) is positive definite.

About the error u−un$u-u_n$

Lemma 3 (Orthogonality of error for elliptic problems) Let u∈V$u\in V$ be the exact solution of the continuous problem (I ) and un$u_n$ the exact solution of the discrete problem (3).
Then the error en=u−un$e_n=u-u_n$, satisjies

.
Proof:Because Vn⊂V$V_n\subset V$, then

Remark 1 (Geometrical interpretation) If the bilinear form a(.,.)$a(.,.)$ is symmetric, it induces an energetic inner product, then

,

i.e., that the error of the Galerkin approximation en=u−un$e_n=u-u_n$ is orthogonal to the Galerkin subspace Vn$V_n$ in the energetic inner product. Hence the approximate solution u, E V, is an orthogonal projection of the exact solution u$u$ onto the Galerkin subspace Vn$V_n$ in the energetic inner product, and thus it is the nearest element in the space Vn$V_n$ to the exact solution u$u$ in the energy norm,

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FEA

Let Ω⊂Rd$\mathrm{\Omega }\subset R^d$ , where d$d$ is the spatial dimension, be an open bounded set. If the Hilbert space V$V$ consists of functions defined in Ω$\mathrm{\Omega }$ and the Galerkin subspaces Vn$V_n$ comprise piecewise-polynomial functions, the Galerkin method is called the Finite element method (FEM).

Consider the model equation

where f∈L2(Ω)$f\in L^2(\mathrm{\Omega })$, in a bounded interval Ω=(a,b)⊂R$\mathrm{\Omega }=(a,b)\subset R$, equipped with the homogeneous Dirichlet boundary conditions. At the beginning let a1$a_1$ and a0$a_0$ be constants and assume a simple load function of the form f(x)=1$f(x)=1$.

The Galerkin procedure assumes a sequence of finite-dimensional subspaces

Consider a partition

and define the finite element mesh

The open intervals

are called finite elements, and the value
is said to be the mesh diameter.

The piecewise linear basis functions vj$v_j$, satisfy vj(xi)=δij$v_j(x_i)={\delta }_{ij}$.

Then by (3), we can get un$u_n$.