Approximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangle

In this paper, we use a hybrid method based on a variant of Trefftz’s method (TM), in
combination with the usual Boundary Collocation Method (BCM) to find the approximate solution
to a singular, two-dimensional mixed boundary-value problem for Laplace’s equation in a rectangular
sheet with one curved side.
After expressing the solution as a finite linear combination of harmonic trial functions, the usual
BCM is used to enforce the boundary condition on the curved side, while a variant of TM is applied to the three remaining sides. The singularity at one corner of the rectangle is treated via the enrichment of the expansion with a specially built harmonic function which has a singularity at one corner. The procedure ultimately produces a rectangular set of linear algebraic equations, which is solved
by QR factorization method.
Numerical results are presented and discussed, in order to assess the efficiency of the proposed
method.