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A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. (English) Zbl 1198.65193

Summary: We identify and study an LDG-hybridizable Galerkin method, which is not an LDG method, for second-order elliptic problems in several space dimensions with remarkable convergence properties. Unlike all other known discontinuous Galerkin methods using polynomials of degree \( k\geq0\) for both the potential as well as the flux, the order of convergence in \( L^2\) of both unknowns is \( k+1\). Moreover, both the approximate potential as well as its numerical trace superconverge in \( L^2\)-like norms, to suitably chosen projections of the potential, with order \( k+2\). This allows the application of element-by-element postprocessing of the approximate solution which provides an approximation of the potential converging with order \( k+2\) in \( L^2\). The method can be thought to be in between the hybridized version of the Raviart-Thomas and that of the Brezzi-Douglas-Marini mixed methods.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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