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Generalized finite differences using fundamental solutions
Author(s) -
Leitão V. M. A.
Publication year - 2009
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2697
Subject(s) - mathematics , method of fundamental solutions , partial differential equation , boundary value problem , domain (mathematical analysis) , finite element method , grid , mathematical analysis , homogeneous , homogeneous differential equation , poincaré–steklov operator , boundary (topology) , work (physics) , differential equation , boundary element method , free boundary problem , ordinary differential equation , boundary knot method , robin boundary condition , geometry , physics , differential algebraic equation , combinatorics , thermodynamics
It is well known that solutions for linear partial differential equations may be given in terms of fundamental solutions. The fundamental solutions solve the homogeneous equation exactly and are obtained from the solution of the inhomogeneous equation where the inhomogeneous term is described by a Dirac delta distribution. Fundamental solutions are the building blocks of the boundary element method and of the method of fundamental solutions and are traditionally used to build boundary‐only global approximations in the domain of interest. In this work the same characteristic of the fundamental solutions, that of solving the homogeneous equation exactly, is used but not to build a global approximation. On the contrary, local approximations are built in such a manner that it is possible to construct finite difference operators that are free from any form of structured grid. Copyright © 2009 John Wiley & Sons, Ltd.