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.
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Scott, R. Finite element convergence for singular data. Numer. Math. 21, 317–327 (1973). https://doi.org/10.1007/BF01436386
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DOI: https://doi.org/10.1007/BF01436386