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This paper proposes and analyzes an optimal preconditioner for a general linear symmetric positive definite (SPD) system by following the basic idea of the well-known BPX framework. The SPD system arises from a large number of nonstandard finite element methods for diffusion problems, including the well-known hybridized Raviart--Thomas and Brezzi-Douglas-Marini mixed element methods, the hybridized discontinuous Galerkin method, the weak Galerkin method, and the nonconforming Crouzeix--Raviart element method. We prove that the presented preconditioner is optimal, in the sense that the condition number of the preconditioned system is independent of the mesh size. Numerical experiments are provided to confirm the theoretical results.

BPX Preconditioner for Nonstandard Finite Element Methods for Diffusion Problems