Steklov Approximations of Harmonic Boundary Value Problems on Planar Regions

TL;DR

This paper provides error estimates for harmonic function approximations on planar regions using Steklov eigenfunctions, supported by explicit representations and computational results for rectangular regions.

Contribution

It introduces explicit error bounds for harmonic Steklov eigenfunction approximations and demonstrates their effectiveness through computational experiments on rectangles.

Findings

Error estimates for harmonic Steklov approximations derived

Explicit harmonic function representations in terms of Steklov eigenfunctions

Computational validation on rectangular regions with known solutions

Abstract

Error estimates for approximations of harmonic functions on planar regions by subspaces spanned by the first harmonic Steklov eigenfunctions are found. They are based on the explicit representation of harmonic functions in terms of these harmonic Steklov eigenfunctions. When the region is a rectangle of aspect ratio h, some computational results regarding these approximations for problems with known explicit solutions are described.