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Penalty‐combined approaches to the Ritz‐Galerkin and finite element methods for singularity problems of elliptic equations
Author(s) -
Li ZiCai
Publication year - 1992
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.1690080103
Subject(s) - mathematics , finite element method , galerkin method , discontinuous galerkin method , mathematical analysis , sobolev space , gravitational singularity , rate of convergence , ritz method , boundary (topology) , singularity , mixed finite element method , convergence (economics) , boundary value problem , computer science , computer network , channel (broadcasting) , physics , economics , thermodynamics , economic growth
Penlty coupling techniques on an interface boundary, artificial or material, are first presented for combining the Ritz–Galerkin and finite element methods. An optimal convergence rate first is proved in the Sobolev norms. Moreover, a significant coupling strategy, L + 1 = O (|ln h |), between these two methods are derived for the Laplace equation with singularities, where L + 1 is the total number of particular solutions used in the Ritz–Galerkin method, and h is the maximal boundary length of quasiuniform elements used in the linear finite element method. Numreical experiments have been carried out for solving the benchmark model: Motz's problem. Both theoretical analysis and numreical experiments clearly display the importance of penalty‐combined methods is solving elliptic equations with singularities.