Finite element method solution of a boundary value problem for an elliptic equation with a Dirac delta function on the right-hand side
D.N. Romanov1, M.V. Urev1,2
1Novosibirsk State University, Novosibirsk, Russia
2Institute of Computational Mathematics and Mathematical Geophysics of Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russia
Keywords: two-dimensional Poisson equation, singular source term, augmented weak formulation, fractional Sobolev spaces, finite element method, error estimate
Abstract
A numerical solution by the finite element method of a homogeneous Dirichlet boundary value problem for an elliptic equation is examined (using a Poisson equation as an example) in a two-dimensional convex polygonal domain Ω with a singular right-hand side given by the Dirac delta function. A theorem on the existence and uniqueness of a generalized solution in the fractional Sobolev space Hs(Ω), 1/2 < s < 1, is proved. An approach to discrete analysis of the problem using the finite element method is proposed and investigated. The results of numerical experiments for a model problem, obtained using the FreeFem++ software, are presented. They confirm the error estimate of the difference between the discrete and exact solutions derived in the paper.
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