2020/01/15 by Jeremy Hoskins, Hoskins, Jeremy, Manas Rachh +1
Computer Science · Engineering · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Electromagnetic Scattering and Analysis #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2001.05434
openalex publication_date 2020/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
In the present paper we describe a class of algorithms for the solution of\nLaplace's equation on polygonal domains with Neumann boundary conditions. It is\nwell known that in such cases the solutions have singularities near the corners\nwhich poses a challenge for many existing methods. If the boundary data is\nsmooth on each edge of the polygon, then in the vicinity of each corner the\nsolution to the corresponding boundary integral equation has an expansion in\nterms of certain (analytically available) singular powers. Using the known\nbehavior of the solution, universal discretizations have been constructed for\nthe solution of the Dirichlet problem. However, the leading order behavior of\nsolutions to the Neumann problem is O(t\μ) for \μ \∈ (-1/2,0)\ndepending on the angle at the corner (compared to O(C+t\μ) with \μ>1/2\nfor the Dirichlet problem); this presents a significant challenge in the design\nof universal discretizations. Our approach is based on using the discretization\nfor the Dirichlet problem in order to compute a solution in the "weak sense" by\nsolving an adjoint linear system; namely, it can be used to compute inner\nproducts with smooth functions accurately, but it cannot be interpolated.\nFurthermore we present a procedure to obtain accurate solutions arbitrarily\nclose to the corner, by solving a sequence of small local subproblems in the\nvicinity of that corner. The results are illustrated with several numerical\nexamples.\n