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

Paola F. Antonietti, Pierre Matalon, Marco Verani

TL;DR

This work considers the solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order (HHO) methods and proposes an explicit technique to compute the discrete normal derivative of an HHO solution of a Laplacian problem.

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.

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

TL;DR

This work considers the solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order (HHO) methods and proposes an explicit technique to compute the discrete normal derivative of an HHO solution of a Laplacian problem.

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.
Paper Structure (25 sections, 4 theorems, 60 equations, 7 figures, 3 tables)

This paper contains 25 sections, 4 theorems, 60 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The continuous splitting of the biharmonic problem
  3. HHO discretization of the Laplacian problem
  4. Mesh definition and notation
  5. Local and broken polynomial spaces
  6. Discrete hybrid formulation
  7. Local problems and cell unknown recovery operator
  8. Discrete normal derivative
  9. Discrete HHO problem
  10. Bilinear form
  11. Discrete scheme
  12. Preconditioner
  13. Algebraic setting
  14. An approximate, sparse matrix
  15. Computation in practice
  16. ...and 10 more sections

Key Result

Lemma 1

For all $v_{\mathcal{F}_h} \in \mathbb{P}^{k}(\mathcal{F}_{h})$ and $\underline{w}_h \in \underline{U}_h$, it holds that

Figures (7)

  • Figure 1: Illustration of the action of the preconditioner.
  • Figure 2: Convergence of the discrete normal derivative $\partial_{\mathrm{n},h}(\underline{u}_h)$ for a smooth solution.
  • Figure 3: Convergence in $L^2$-norm of the discrete solution $(\omega_h, \psi_h)$ with respect to the exact solution $\psi(x, y) = x\sin(\pi y)e^{-xy}$. Square domain discretized by Cartesian meshes.
  • Figure 4: Convergence in $L^2$-norm of the discrete solution $(\omega_h, \psi_h)$ with respect to the exact solution $\psi(x, y) = x\sin(\pi y)e^{-xy}$. Square domain discretized by polygonal meshes.
  • Figure 5: Convergence in $L^2$-norm of the discrete solution $(\omega_h, \psi_h)$ with respect to the exact solution $\psi(x, y) = x^4(x - 1)^2 y^4(y - 1)^2$. Square domain discretized by Cartesian meshes.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • ...and 2 more