Iterative solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order method

TL;DR

This paper introduces an iterative hybrid high-order method for solving the biharmonic equation in mixed form, featuring a new boundary unknown, an explicit normal derivative computation, and a preconditioner to enhance convergence, validated through numerical tests.

Contribution

It develops a specialized iterative scheme for the HHO discretization of the biharmonic equation, including a novel normal derivative computation and an effective preconditioner.

Findings

The iterative scheme converges efficiently in 2D and 3D cases.

The explicit normal derivative computation improves accuracy.

The preconditioner significantly accelerates Krylov method convergence.

Abstract

We consider the solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order (HHO) methods. The two resulting second-order elliptic problems can be decoupled via the introduction of a new unknown, corresponding to the boundary value of the solution of the first Laplacian problem. This technique yields a global linear problem that can be solved iteratively via a Krylov-type method. More precisely, at each iteration of the scheme, two second-order elliptic problems have to be solved, and a normal derivative on the boundary has to be computed. In this work, we specialize this scheme for the HHO discretization. To this aim, an explicit technique to compute the discrete normal derivative of an HHO solution of a Laplacian problem is proposed. Moreover, we show that the resulting discrete scheme is well-posed. Finally, a new preconditioner is designed to speed up the convergence of the Krylov method. Numerical experiments assessing the performance of the proposed iterative algorithm on both two- and three-dimensional test cases are presented.