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.

翻译：考虑采用混合高阶方法离散化的混合形式双调和方程求解问题。通过引入与第一个拉普拉斯问题解边界值对应的新未知量，可将所得两个二阶椭圆问题解耦。该技术产生一个全局线性问题，可通过Krylov型方法迭代求解。具体而言，在该格式的每次迭代中，需求解两个二阶椭圆问题，并计算边界上的法向导数。本文将该格式专门用于混合高阶(HHO)离散化。为此，提出了计算拉普拉斯问题HHO解离散法向导数的显式技术。同时证明了所得离散格式的适定性。最后设计了新的预条件器以加速Krylov方法的收敛。给出了二维和三维测试算例中迭代算法性能的数值实验评估。