Towards computing high-order p-harmonic descent directions and their limits in shape optimization

TL;DR

This paper extends an algorithm for the scalar p-Laplace problem to the vector-valued case in shape optimization, enabling efficient computation of high-order p-harmonic descent directions without p-iteration.

Contribution

It introduces a novel method for vector-valued p-Laplacian problems that avoids p-iteration, with polynomial convergence and validation through shape deformation experiments.

Findings

Efficient computation of high-order p-harmonic solutions.

Polynomial convergence of Newton iterations.

Insights into limit behaviors and sign change challenges.

Abstract

We present an extension of an algorithm for the classical scalar $p$-Laplace Dirichlet problem to the vector-valued $p$-Laplacian with mixed boundary conditions in order to solve problems occurring in shape optimization using a $p$-harmonic approach. The main advantage of the proposed method is that no iteration over the order $p$ is required and thus allow the efficient computation of solutions for higher orders. We show that the required number of Newton iterations remains polynomial with respect to the number of grid points and validate the results by numerical experiments considering the deformation of shapes. Further, we discuss challenges arising when considering the limit of these problems from an analytical and numerical perspective, especially with respect to a change of sign in the source term.