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We present an extension of an algorithm for the classical scalar $p$-LaplaceDirichlet problem to the vector-valued $p$-Laplacian with mixed boundaryconditions in order to solve problems occurring in shape optimization using a$p$-harmonic approach. The main advantage of the proposed method is that noiteration over the order $p$ is required and thus allow the efficientcomputation of solutions for higher orders. We show that the required number ofNewton iterations remains polynomial with respect to the number of grid pointsand validate the results by numerical experiments considering the deformationof shapes. Further, we discuss challenges arising when considering the limit ofthese problems from an analytical and numerical perspective, especially withrespect to a change of sign in the source term.

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