A Highly Accurate Solver for the Mixed-Boundary Potential Problem and Singular Problem in Arbitrary Plane Domain

A highly accurate new solver is de- veloped to deal with interior and exterior mixed- boundary value problems for two-dimensional Laplace equation, including the singular ones. To promote the present study, we introduce a circu- lar artificial boundary which is uniquely deter- mined by the physical problem domain, and de- rive a Dirichlet to Robin mapping on that arti- ficial circle, which is an exact boundary condi- tion described by the first kind Fredholm integral equation. As a consequence, we obtain a modi- fied Trefftz method equipped with a characteristic length factor, ensuring that the new solver is sta- ble because the condition number can be greatly reduced. Then, the collocation method is used to derive a linear equations system to determine the Fourier coefficients. We find that the new method is powerful even for the problem with complex boundary shape and with adding random noise on the boundary data. It is also applicable to the sin- gular problem of Motz type, resulting to a highly accurate result never seen before. Keyword: Laplace equation, Artificial circle, DtR mapping, Modified Trefftz method, Mixed- boundary value problem, Singular problem

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