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：我们考虑由混合高阶（HHO）方法离散化的混合形式双调和方程的求解问题。通过引入一个新未知量（对应于第一个拉普拉斯问题解的边值），可将由此产生的两个二阶椭圆问题解耦。该技术得到一个全局线性问题，可通过Krylov型方法迭代求解。更具体地说，在该格式的每次迭代中，需要求解两个二阶椭圆问题，并计算边界上的法向导数。本文针对HHO离散化对该格式进行专门研究。为此，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，我们证明所得到的离散格式是适定的。最后，设计了一个新的预条件器以加速Krylov方法的收敛。文中还给出了在二维和三维测试案例中评估所提迭代算法性能的数值实验。