Pavarino proved that the additive Schwarz method with vertex patches and a low-order coarse space gives a $p$-robust solver for symmetric and coercive problems. However, for very high polynomial degree it is not feasible to assemble or factorize the matrices for each patch. In this work we introduce a direct solver for separable patch problems that scales to very high polynomial degree on tensor product cells. The solver constructs a tensor product basis that diagonalizes the blocks in the stiffness matrix for the internal degrees of freedom of each individual cell. As a result, the non-zero structure of the cell matrices is that of the graph connecting internal degrees of freedom to their projection onto the facets. In the new basis, the patch problem is as sparse as a low-order finite difference discretization, while having a sparser Cholesky factorization. We can thus afford to assemble and factorize the matrices for the vertex-patch problems, even for very high polynomial degree. In the non-separable case, the method can be applied as a preconditioner by approximating the problem with a separable surrogate. We demonstrate the approach by solving the Poisson equation and a $H(\mathrm{div})$-conforming interior penalty discretization of linear elasticity in three dimensions at $p = 15$.

翻译：Pavarino证明了带有顶点补丁和低阶粗空间的加法Schwarz方法对于对称和强制问题具有$p$-鲁棒性。然而，对于非常高次多项式，组装或分解每个补丁的矩阵并不可行。在本工作中，我们引入了一种针对可分离补丁问题的直接求解器，该求解器可扩展到张量积单元上的极高多项式次数。该求解器构建了一个张量积基，对角化每个单元内部自由度刚度矩阵中的块。因此，单元矩阵的非零结构对应于连接内部自由度与其在面上的投影的图。在新基下，补丁问题的稀疏性与低阶有限差分离散化相当，同时具有更稀疏的Cholesky分解。因此，即使对于非常高次多项式，我们也能负担得起组装和分解顶点补丁问题的矩阵。在不可分离情况下，该方法可通过用可分离近似替代问题而作为预条件子应用。我们通过求解三维中$p=15$时的泊松方程和$H(\mathrm{div})$-一致内部罚离散线性弹性问题来展示该方法。