Finite element convergence for singular data

AbstractConvergence of the finite element solutionuh 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 outsideH1, 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 inL2 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 inH5/2, but as with the approximation to the Green's function, we prove the full rate of convergence:‖u−uh‖1=O (h8/2) with, say, piecewise quadratics.