The Green function of the Poisson equation in two dimensions is not contained in the Sobolev space $H^1(\Omega)$ 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 $L^2$-error estimates of almost optimal order are shown. As a byproduct, we show for the Poisson equation with a right-hand side in $L^2$ that appropriate mesh refinement near some interior point diminishes the error at this point by nearly one order.

MSC codes

  1. 65N30
  2. 65N15


  1. Dirac measure
  2. Green function
  3. fundamental solution
  4. finite element method
  5. graded mesh
  6. error estimate

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Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 992 - 1005
ISSN (online): 1095-7170


Submitted: 23 November 2009
Accepted: 16 February 2011
Published online: 19 May 2011

MSC codes

  1. 65N30
  2. 65N15


  1. Dirac measure
  2. Green function
  3. fundamental solution
  4. finite element method
  5. graded mesh
  6. error estimate



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