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A highly accurate collocation Trefftz method for solving the Laplace equation in the doubly connected domains
Author(s) -
Liu CheinShan
Publication year - 2008
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.20257
Subject(s) - mathematics , solver , laplace's equation , dirichlet problem , partial differential equation , laplace transform , domain (mathematical analysis) , boundary value problem , collocation method , collocation (remote sensing) , mathematical analysis , dirichlet boundary condition , singularity , integral equation , boundary (topology) , dirichlet distribution , ordinary differential equation , differential equation , mathematical optimization , computer science , machine learning
A highly accurate new solver is developed to deal with the Dirichlet problems for the 2D Laplace equation in the doubly connected domains. We introduce two circular artificial boundaries determined uniquely by the physical problem domain, and derive a Dirichlet to Dirichlet mapping on these two circles, which are exact boundary conditions described by the first kind Fredholm integral equations. As a direct result, we obtain a modified Trefftz method equipped with two characteristic length factors, ensuring that the new solver is stable because the condition number can be greatly reduced. Then, the collocation method is used to derive a linear equations system to determine the unknown coefficients. The new method possesses several advantages: mesh‐free, singularity‐free, non‐illposedness, semi‐analyticity of solution, efficiency, accuracy, and stability. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007