Abstract

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

Keywords

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

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References

1.
T. Apel, A.-M. Sändig, and J. R. Whiteman, Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains, Math. Methods Appl. Sci., 19 (1996), pp. 63–85.
2.
R. Araya, E. Behrens, and R. Rodríguez, A posteriori error estimates for elliptic problems with Dirac delta source terms, Numer. Math., 105 (2006), pp. 193–216.
3.
I. Babuška, Error-bounds for finite element method, Numer. Math., 16 (1971), pp. 322–333.
4.
R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10 (2001), pp. 1–102.
5.
O. Benedix and B. Vexler, A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints, Comput. Optim. Appl., 44 (2009), pp. 3–25.
6.
E. Casas, $L^2$ estimates for the finite element method for the Dirichlet problem with singular data, Numer. Math., 47 (1985), pp. 627–632.
7.
E. Casas, Control of an elliptic problem with pointwise state constraints, SIAM J. Control Optim., 24 (1986), pp. 1309–1318.
8.
S. Du and X. Xie, Residual-based a posteriori error estimates of non-conforming finite element method for elliptic problems with Dirac delta source terms, Sci. China Ser. A, 51 (2008), pp. 1440–1460.
9.
K. Eriksson, Finite element methods of optimal order for problems with singular data, Math. Comp., 44 (1985), pp. 345–360.
10.
K. Eriksson, Improved accuracy by adapted mesh-refinements in the finite element method, Math. Comp., 44 (1985), pp. 321–343.
11.
J. Frehse and R. Rannacher, Eine $L^1$-Fehlerabschätzung für diskrete Grundlösungen in der Methode der finiten Elemente, in Finite Elemente (Tagungsband des Sonderforschungsbereichs 72), J. Frehse, R. Leis, and R. Schaback, eds., Bonn. Math. Schrift. 89, Inst. Angew. Math., University of Bonn, Bonn, Germany, 1976, pp. 92–114.
12.
R. Fritzsch, Optimale Finite-Elemente-Approximationen für Funktionen mit Singularitäten, Ph.D. thesis, TU Dresden, Dresden, Germany, 1990.
13.
A. Günther and M. Hinze, A posteriori error control of a state constrained elliptic control problem, J. Numer. Math., 16 (2008), pp. 335–350.
14.
M. Hintermüller and R. H. W. Hoppe, Goal-oriented adaptivity in pointwise state constrained optimal control of partial differential equations, SIAM J. Control Optim., 48 (2010), pp. 5468–5487.
15.
V. John and G. Matthies, MooNMD—a program package based on mapped finite element methods, Comput. Vis. Sci., 6 (2004), pp. 163–169.
16.
H. Li, A. Mazzucato, and V. Nistor, Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains, Electron. Trans. Numer. Anal., 37 (2010), pp. 41–69.
17.
L. A. Oganesyan and L. A. Rukhovets, Variational-difference schemes for linear second-order elliptic equations in a two-dimensional region with piecewise smooth boundary, Zh. Vychisl. Mat. Mat. Fiz., 8 (1968), pp. 97–114 in Russian; USSR Comput. Math. and Math. Phys., 8 (1968), pp. 129–152 (in English).
18.
R. Rannacher, private communication.
19.
R. Rannacher and B. Vexler, A priori error estimates for the finite element discretization of elliptic parameter identification problems with pointwise measurements, SIAM J. Control Optim., 44 (2005), pp. 1844–1863.
20.
A. H. Schatz and L. B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains II. Refinements, Math. Comp., 33 (1979), pp. 465–492.
21.
R. Scott, Finite element convergence for singular data, Numer. Math., 21 (1973), pp. 317–327.

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Published In

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

History

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

MSC codes

  1. 65N30
  2. 65N15

Keywords

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

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