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Highly Accurate Numerical Solutions for Potential Problem and Singular Problem in Arbitrary Plane Domain

In: Computational Mechanics

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  • Chein-Shan Liu

    (Taiwan Ocean University, Department of Mechanical and Mechatronic Engineering)

Abstract

A highly accurate new solver is developed to deal with the interior and exterior mixed-boundary value problems for the 2D Laplace equation, including the singular one. To motivate the present study, we introduce a circular artificial boundary which is uniquely determined by the physical problem domain, and derive a Dirichlet to Robin mapping on that circle, which is an exact boundary condition described by the first kind Fredholm integral equation. As a direct result, we obtain a modified Trefftz method equipped with a characteristic length factor, which ensures 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 Fourier coeficients. We find that the new method is powerful even for the problem with very complex boundary shape and with adding random noise on the boundary data. It is also applicable to the computation of singular problem of Motz type, resulting to a high accuracy never seen before.

Suggested Citation

  • Chein-Shan Liu, 2007. "Highly Accurate Numerical Solutions for Potential Problem and Singular Problem in Arbitrary Plane Domain," Springer Books, in: Computational Mechanics, pages 375-375, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-75999-7_175
    DOI: 10.1007/978-3-540-75999-7_175
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