Finite element convergence for singular data

Convergence of the finite element solutionu ^h of the Dirichlet problem u= is proved, where is the Dirac -function (unit impulse). In two dimensions, the Green's function (fundamental solution)u lies outsideH ¹, but we are able to prove that . Since the singularity ofu is logarithmic, we conclude that in two dimensions the function log can be approximated inL ² near the origin by piecewise linear functions with an errorO (h). We also consider the Dirichlet problem u=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.