This paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection-diffusion-reaction problems discretized by Galerkin or discontinuous Galerkin methods. We expand on the approach introduced by Gergelits et al. and adapt it to the more general settings, assuming that both the original and preconditioning matrices are composed of sparse matrices of very low ranks, representing local contributions to the global matrices. When applied to a symmetric problem, the method provides bounds to all individual eigenvalues of the preconditioned matrix. We show that this preconditioning strategy works not only for Galerkin discretization, but also for the discontinuous Galerkin discretization, where local contributions are associated with individual edges of the triangulation. In the case of non-symmetric problems, the method yields guaranteed bounds to real and imaginary parts of the resulting eigenvalues. We include some numerical experiments illustrating the method and its implementation, showcasing its effectiveness for the two variants of discretized (convection-)diffusion-reaction problems.

翻译：本文聚焦于线性二阶偏微分方程（包括由Galerkin或不连续Galerkin方法离散的对流-扩散-反应问题）的预处理新概念的设计、分析与实现。我们拓展了Gergelits等人提出的方法，并使其适用于更一般的场景，假设原始矩阵和预处理矩阵均由代表全局矩阵局部贡献的极低秩稀疏矩阵组成。当应用于对称问题时，该方法可为预处理矩阵的所有特征值提供界限。我们证明该预处理策略不仅适用于Galerkin离散化，也适用于不连续Galerkin离散化，其中局部贡献与三角剖分的各边相关联。针对非对称问题，该方法可为所得特征值的实部和虚部提供可保证的界限。我们通过数值实验说明该方法及其实现，展示了其在两类离散化（对流-）扩散-反应问题中的有效性。