Integral transform solution of eigenvalue problems within irregular geometries: Comparative analysis of different methodologies

Abstract

Integral transform solutions for differential eigenvalue problems within irregular geometries using different methodologies have been formally developed and comparatively analyzed. The first method, termed the Coincident Domain Approach (CDA) is based on the solution of the eigenvalue problem employing the original problem domain, while the second, coined the Fictitious Domain Approach (FDA), involves redefining the solution domain using a fictitious, yet regular, region that encompasses the original problem boundaries. After presenting series solutions for a general problem using the two approaches, 2D test-case problems — with Dirichlet and Neumann boundary conditions at the irregular boundaries — were selected for comparing both methodologies from a computational standpoint. A thorough error analysis of the numerical results was presented, showing that the CDA greatly outperforms the FDA for the Dirichlet case; on the other hand, the opposite trend was seen for the Neumann case, for which the FDA clearly presented better results.

Additional information

Funding

The authors would like to acknowledge the financial support provided by the Brazilian Government Funding Agencies, CAPES, CNPq, and FAPERJ.