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：本文研究混合形式双调和方程经杂交高阶（HHO）方法离散后的求解问题。通过引入对应于第一个拉普拉斯问题解边界值的新未知量，可将两个导出的二阶椭圆问题解耦。该技术产生了一个可通过Krylov型方法迭代求解的全局线性问题。更精确地说，在该方案的每次迭代中，需要求解两个二阶椭圆问题并计算边界上的法向导数。本工作针对HHO离散化专门设计了该求解方案。为此，我们提出了一种显式技术来计算拉普拉斯问题HHO解的离散法向导数。此外，我们证明了所得离散格式是适定的。最后，设计了一种新的预条件子以加速Krylov方法的收敛。数值实验展示了所提迭代算法在二维和三维测试案例中的性能表现。