A Priori Mesh Grading for an Elliptic Problem with Dirac Right-Hand Side

The Green function of the Poisson equation in two dimensions is not contained in the Sobolev space H1(Ω) such that finite element error estimates for the discretization of a problem with the Dirac measure on the right hand-side are nonstandard and quasi-uniform meshes are inappropriate. By using graded meshes L2-error estimates of almost optimal order are shown. As a byproduct, we show for the Poisson equation with a right-hand side in L2 that appropriate mesh refinement near some interior point diminishes the error at this point by nearly one order. Key words. Dirac measure, Green function, fundamental solution, finite element method, graded mesh, error estimate AMS subject classifications. 65N30, 65N15 DOI. 10.1137/090778018

• Type: Journal
• Year: 2011
• Where: SIAMNUM
• Authors: Thomas Apel, Olaf Benedix, Dieter Sirch, Boris Vexler
• Keywords: Element Method, Error Estimates, SIAMNUM 2011, Uniform Meshes