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The research aims to develop a spectral method for solving elliptic partial differential equations on two-dimensional domains using harmonic maps.

A Spectral Method with Harmonic Map for Elliptic PDEs on General Two-Dimensional Domains

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The research aims to develop a spectral method for solving elliptic partial differential equations on two-dimensional domains using harmonic maps.
Development of a Fourier-Legendre spectral-Galerkin method
Application of harmonic maps for geometrically complex domains
Mathematical proof of optimal convergence in H^1-norm
The method shows optimal convergence regardless of boundary smoothness
Demonstrated effectiveness on star-shaped and nonstar-shaped domains
Validated with numerical examples across various complex geometries

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Demonstrates a new method to solve elliptic PDEs in complex geometries, indicating wide applicability.

Shi et al. (2026) studied this question.

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