Infinite element approximation to axial symmetric Stokes flow
Journal of Computational Mathematics
On the numerical treatment of boundary singularities in elliptic problems
Journal of Computational Physics
On the multi-level splitting of finite element spaces
Numerische Mathematik
SIAM Journal on Numerical Analysis
Computer Methods in Applied Mechanics and Engineering
Mathematics of Computation
A spectral Galerkin method for a boundary integral equation
Mathematics of Computation
Numerical solution for exterior problems
Numerische Mathematik
On the coupling boundary integral and finite element methods for the exterior Stokes problem in 3-D
SIAM Journal on Numerical Analysis
Stress intensity factors and improved convergence estimates at a corner
SIAM Journal on Numerical Analysis
Boundary methods for solving elliptic problems with singularities and interfaces
SIAM Journal on Numerical Analysis
The optimal convergence rate of the p-version of the finite element method
SIAM Journal on Numerical Analysis
On the boundary element method with mesh refinement on curves with corners
Journal of Computational and Applied Mathematics
Finite element eigenvalues for the Laplacian over an L-shaped domain
Journal of Computational Physics
On semi-infinite spectral elements for Poisson problems with re-entrant boundary singularities
Journal of Computational and Applied Mathematics
Journal of Computational Physics
Trefftz method: elastic and elasto-plastic problems
Computer Methods in Applied Mechanics and Engineering
An efficient method for subtracting off singularities at corners for Laplace's equation
Journal of Computational Physics
A nonconforming combination for solving elliptic problems with interfaces
Journal of Computational Physics
Computer Methods in Applied Mechanics and Engineering
On the conditioning of finite element equations with highly refined meshes
SIAM Journal on Numerical Analysis
Regularity and numerical solution of eigenvalue problems with piecewise analytic data
SIAM Journal on Numerical Analysis
Adaptive domain extension and adaptive grids for unbounded spherical elliptic PDEs
SIAM Journal on Scientific and Statistical Computing
The simplified hybrid-combined methods for Laplace's equation with singularities
Journal of Computational and Applied Mathematics
Numerical conformal mapping onto a rectangle with applications to the solution of Laplacian problems
Journal of Computational and Applied Mathematics
Coupling of finite and boundary element methods for an elastoplastic interface problem
SIAM Journal on Numerical Analysis
Boundary and interface conditions within a finite element preconditioner for spectral methods
Journal of Computational Physics
The p- and h-p version of the finite element method, an overview
ICOSAHOM '89 Proceedings of the conference on Spectral and high order methods for partial differential equations
The h-p version of the boundary element method with geometric mesh on polygonal domains
ICOSAHOM '89 Proceedings of the conference on Spectral and high order methods for partial differential equations
SIAM Journal on Numerical Analysis
Finite element methods for the Helmholtz equation in an exterior domain: model problems
Computer Methods in Applied Mechanics and Engineering
An efficient finite element method for treating singularities in Laplace's equation
Journal of Computational Physics
A domain decomposition preconditioner with reduced rank interdomain coupling
Applied Numerical Mathematics - II on Domain decomposition; Guest Editor: W. Proskurowski
Mathematica: a system for doing mathematics by computer (2nd ed.)
Mathematica: a system for doing mathematics by computer (2nd ed.)
Domain decomposition with local mesh refinement
SIAM Journal on Scientific and Statistical Computing
The p-version of the finite element method for the elliptic boundary value problems with interfaces
Computer Methods in Applied Mechanics and Engineering
Computer Methods in Applied Mechanics and Engineering
On the finite element method for a singular problem
Applied Mathematics and Computation
Journal of Computational Physics
Coupling of finite elements and boundary elements for some nonlinear interface problems
Computer Methods in Applied Mechanics and Engineering - Special issue on reliability in computational mechanics
The Schwarz alternating method for singularity problems
SIAM Journal on Scientific Computing
Solution of Helmholtz equation in the exterior domain by elementary boundary integral methods
Journal of Computational Physics
On the coupling of boundary integral and mixed finite element methods
Journal of Computational and Applied Mathematics
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Coupling of FEM and BEM for interface problems in viscoplasticity and plasticity with hardening
SIAM Journal on Numerical Analysis
Eigenmodes of Isospectral Drums
SIAM Review
A Finite-Element Method for Laplace- and Helmholtz-Type Boundary Value Problems with Singularities
SIAM Journal on Numerical Analysis
Penalty combinations of the Ritz-Galerkin and finite difference methods for singularity problems
Journal of Computational and Applied Mathematics
Global superconvergence in combinations of Ritz-Galerkin-FEM for singularity problems
Journal of Computational and Applied Mathematics
The Numerical Solution of the Integral Equation Formulation of the Single Interface Neumann Problem
SIAM Journal on Numerical Analysis
Computers & Mathematics with Applications
Computers & Mathematics with Applications
Journal of Computational and Applied Mathematics
Hi-index | 0.98 |
This paper provides a survey for treatments for singularity problems of elliptic equations. We take the Laplace equation on polygons as an example, and choose Motz's problem as a benchmark of singularity problems. The conformal transformation method of Whiteman and Papamichael [1] and Rosser and Papamichael [2] is the most accurate method for seeking the leading coefficients of the solution expansions of Motz's problem. In this paper, we provide the first 100 leading coefficients D"0-D"9"9, where d"0 and D"9"9 have 199 and 89 significant decimal digits, respectively, which serve as the test coefficients for other numerical methods. Other treatments may be classified into three categories: local refinements, singular function method of Fix [3], Wigley [4], and Blum and Dobrowolski [5], and the combined methods. The local refinements have a wide range of applications due to less requirements; the combined methods are most efficient because the singular functions may be chosen to fit best the singularity of the solutions. Also, the natural boundary element method may handle well the unbounded domain problems. Moreover, the combined methods are introduced briefly with the coupling techniques, when other numerical methods are interpreted as a kind of finite element methods by using different admissible functions and different integration rules. Different coupling techniques are, indeed, intended to remedy the nonconformality of the admissible functions along the common (interior) boundary of different numerical methods used. Besides, about 300 publications related to singularities and their treatments are collected in this paper.