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.

中文翻译：

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混合高阶法离散混合形式双调和方程的迭代解


我们考虑用混合高阶 (HHO) 方法离散化的混合形式的双调和方程的解。由此产生的两个二阶椭圆问题可以通过引入一个新的未知数来解耦，该未知数对应于第一个拉普拉斯问题的解的边界值。该技术产生了一个全局线性问题，可以通过 Krylov 型方法迭代解决。更准确地说，在该方案的每次迭代中，必须解决两个二阶椭圆问题，并且必须计算边界上的法态导数。在这项工作中，我们专门将该方案用于 HHO 离散化。为此，提出了一种计算拉普拉斯问题 HHO 解的离散法态导数的显式技术。此外，我们表明所得到的离散方案是适定的。最后，设计了一种新的预处理器来加速Krylov方法的收敛。提出了评估所提出的迭代算法在二维和三维测试用例上的性能的数值实验。