Computational analysis demonstrates layer potential representations on finitely connected tori, indicating faster spectral convergence for boundary value problems than existing approaches.
Key Points
To develop and analyze layer potential methods for representing harmonic functions on finitely connected tori without requiring lattice sums of free-space Green's functions.
Formulated layer potentials using a doubly periodic, nonharmonic Green's function expressed via Jacobi theta functions or modified Weierstrass sigma functions.
Proved compactness and boundary limit properties of single- and double-layer potential operators, explicitly constructing null spaces for boundaries with multiple connected components.
Implemented Nyström discretizations to compute solutions for Dirichlet, Neumann, and Steklov eigenvalue problems on domains with irregularly shaped holes.
Proved that the single- and double-layer potential operators are compact linear operators with boundary limit properties matching Euclidean domains.
Constructed explicit representations for the nontrivial null space of the Fredholm operator of the second kind associated with multi-component boundaries.
Demonstrated spectral convergence in numerical experiments, yielding faster convergence rates than the method of particular solutions for tori with irregular geometries.