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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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