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. Romanov^a, M. V. Urev<sup>ab

a</sup> Novosibirsk State University, Russia
^b Institute of Computational Mathematics and Mathematical Geophysics of Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russia

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 $\Omega$ 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 $H^s(\Omega)$, $1/2 < s <1$, 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.

Key words: two-dimensional Poisson equation, singular source term, augmented weak formulation, fractional Sobolev spaces, finite element method, error estimate.

UDC: 519.632

Received: 28.03.2025
Revised: 12.05.2025
Accepted: 16.06.2025

DOI: 10.15372/SJNM20250403