0
$\begingroup$

Consider the finite element method for the approximate solution of the equation $$ \nabla\cdot(a(x)\nabla u(x))=f(x) \text{ (in some domain)} $$ or, in weak form (assuming the right boundary conditions): Find $u\in H^1$ such that $$ \int a(x) \nabla u(x)\nabla v(x)=\int f(x) v(x)\quad\forall v\in H^1 $$

The finite element method replaces $u\in H^1$ and $v\in H^1$ by $u\in V$ and $v\in V$ for some finite dimensional vector space $V$ with a basis $(\psi_i)_{i=1}^n$ of compactly supported piecewise polynomials. This reduces the problem to a finite-dimensional linear system $$ Ax=b $$ where the entries of $A$ are $a_{ij}=\int a(x)\nabla \psi_i(x)\nabla\psi_j(x)$. When $a(x)$ is constant, these integrals can usually be computed exactly.

My question: When $a(x)$ is nonconstant, but $a\in C^m$, how exact do I need to approximate the entries $a_{ij}$? For example, if I use piecewise linear finite elements, and I want to maintain the convergence rate $\|u_h-u\|_{L^2}\leq Ch^2$ that exact computation of the $a_{ij}$ would provide, does it suffice to approximate the integrals by a midpoint rule?

$\endgroup$
0
$\begingroup$

There is a result in the Ciarlet book (Thm 4.1.6), which says that if the space contains polynomials of degree $k$, the numerical integration is exact for polynomials of degree $2k-2$, then we have $$ \|\nabla u-\nabla u_h\|_{L^2(\Omega)} \le c \ h^k \ ( \|a\|_{W^{k,\infty}}\|u\|_{H^{k+1}(\Omega)} + \|f\|_{W^{k,q}(\Omega)}) $$ where $q>2$, $k>n/q$.

Hence, for piecewise linear polynomials the midpoint rule is enough. I am not sure, whether this is sufficient to get the optimal rate for the $L^2$-error.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.