Regularity and finite element approximation for two-dimensional elliptic equations with line Dirac sources

by   Hengguang Li, et al.

We study the elliptic equation with a line Dirac delta function as the source term subject to the Dirichlet boundary condition in a two-dimensional domain. Such a line Dirac measure causes different types of solution singularities in the neighborhood of the line fracture. We establish new regularity results for the solution in a class of weighted Sobolev spaces and propose finite element algorithms that approximate the singular solution at the optimal convergence rate. Numerical tests are presented to justify the theoretical findings.



page 16

page 17

page 18

page 19


An adaptive finite element method for two-dimensional elliptic equations with line Dirac sources

In this paper, we study an adaptive finite element method for the ellipt...

Optimal finite elements for ergodic stochastic two-scale elliptic equations

We develop an essentially optimal finite element approach for solving er...

Convergence of a regularized finite element discretization of the two-dimensional Monge-Ampère equation

This paper proposes a regularization of the Monge-Ampère equation in pla...

Finite Element Methods for Isotropic Isaacs Equations with Viscosity and Strong Dirichlet Boundary Conditions

We study monotone P1 finite element methods on unstructured meshes for f...

Coupled Flow and Mechanics in a 3D Porous Media with Line Sources

In this paper, we consider the numerical approximation of the quasi-stat...

Adaptive finite element approximations for elliptic problems using regularized forcing data

We propose an adaptive finite element algorithm to approximate solutions...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.