Finite element convergence for singular data

Abstract

Convergence of the finite element solutionu ^h of the Dirichlet problem Δu=δ is proved, where δ is the Dirac δ-function (unit impulse). In two dimensions, the Green's function (fundamental solution)u lies outsideH ¹, but we are able to prove that$\parallel u - u^h \parallel _{L^2 } = O (h)$. Since the singularity ofu is logarithmic, we conclude that in two dimensions the function log γ can be approximated inL ² near the origin by piecewise linear functions with an errorO (h). We also consider the Dirichlet problem Δu=f, wheref is piecewise smooth but discontinuous along some curve. In this case,u just fails to be inH ^5/2, but as with the approximation to the Green's function, we prove the full rate of convergence:‖u−u ^h‖₁=O (h ^8/2) with, say, piecewise quadratics.