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$$L^ 2$$ estimates for the finite element method for the Dirichlet problem with singular data. (English) Zbl 0561.65071
We consider the approximation by the finite element method of second order elliptic problems on convex domains and with homogeneous Dirichlet condition on the boundary. In these problems the data are Borel measures. Using a quasiuniform mesh of finite elements and polynomials of degree $$\leq 1$$, we prove that in two dimensions the convergence is of order h in $$L^ 2$$ and in three dimensions of order $$h^{1/2}$$.

MSC:
 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
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References:
 [1] Adams, R.A.: Sobolev spaces. New York: Academic Press 1975 · Zbl 0314.46030 [2] Babu?ka, I.: Error-bounds for the finite element method. Numer. Math.16, 322-333 (1971) · Zbl 0214.42001 · doi:10.1007/BF02165003 [3] Babu?ka, I., Aziz, A.K.: Survey lectures on the mathematical foundations of the finite element method. Section 6.3.6, Proc. of the Conf. on the Mathematical Foundations on the Finite Element Method. New York: Academic Press 1973 [4] Bonnans, J.F., Casas, E.: Contrôle de systèmes non lineaires comportant des contraintes distribuées sur l’état. Rapport INRIA no 300 (1984) [5] Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optimization (Submitted) · Zbl 0606.49017 [6] Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam: North-Holland 1978 · Zbl 0383.65058 [7] Grisvard, P.: Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain. In: Numerical Solution of Partial Differential Equations III (Sysnspade 1975) (B. Hubbard, ed.), pp. 207-274. New York: Academic Press 1976 [8] Ne?as, J.: Les méthodes directes en théorie des equations elliptiques. Paris: Masson 1967 [9] Rudin, W.: Real and complex analysis. New York: McGraw-Hill 1966 · Zbl 0142.01701 [10] Scott, R.: Finite element convergence for singular data. Numer. Math.21, 317-327 (1973) · Zbl 0255.65037 · doi:10.1007/BF01436386 [11] Scott, R.: OptimalL ? estimates for the finite element method on irregular meshes. Math. Comput.30, 681-697 (1976) · Zbl 0349.65060 [12] Stampacchia, G.: Le problème de Dirichlet pour les equations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier, Grenoble15, 189-258 (1965) · Zbl 0151.15401
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