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An approach combining the Ritz‐Galerkin and finite difference methods
Author(s) -
Li ZiCai
Publication year - 1989
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.1690050402
Subject(s) - mathematics , finite element method , galerkin method , discontinuous galerkin method , rate of convergence , ritz method , convergence (economics) , mathematical analysis , finite difference , boundary value problem , boundary (topology) , gravitational singularity , finite difference method , channel (broadcasting) , physics , economic growth , electrical engineering , economics , thermodynamics , engineering
The nonconforming combination of Ritz‐Galerkin and finite difference methods is presented for solving elliptic boundary value problems with singularities. The Ritz‐Galerkin method is used in the subdomains including singularities, the finite difference method is used in the rest of the solution domain. Moreover, on the common boundary of two regions where two different methods are used, the continuity conditions are constrained only on the nodes of difference grids. Theoretical analysis and numerical experiments have shown that average errors of numerical solutions and their generalized derivatives can reach the convergence rate O ( h 2‐δ ), where h is the mesh spacing of uniform difference grids, and δ is an arbitrarily small, positive number. This convergence rate is better than O ( h ), obtained by the nonconforming combination of the Ritz‐Galerkin and finite element methods.