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An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations. (English) Zbl 1187.65110

The authors are concerned with the numerical solution of steady and time-dependent linear convection-diffusion equations. They present an hybridizable discontinuous Galerkin method that significantly reduce the number of globally coupled degrees of freedom in the discontinuous Galerkin approximations. When the problem is time dependent a backward difference formula is used for the discretization of the time derivative.
This results in a high order method in space and time: when using polynomials of degree \(p \geq 0\) for the space discretization and a time discretization scheme of order \(p+1\), the order of convergence in the \(L^2\)-norm of the scalar variable and the flux is \(p+1\). An element-by-element postprocessing scheme is proposed leading to new approximations of the scalar variable and the flux that converge with order \(p+2\) and \(p+1\) respectively in the \(L^2\)-norm. Several numerical examples are presented to show the accuracy and stability of the proposed scheme in a wide range of regimes.

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
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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