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An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations. (English) Zbl 1177.65150

Summary: We present hybridizable discontinuous Galerkin methods for the numerical solution of steady and time-dependent nonlinear convection-diffusion equations. The methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary. Applying the Newton-Raphson procedure and the hybridization technique, we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every Newton iteration. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced.
We then extend the method to time-dependent problems by approximating the time derivative by means of backward difference formulae. When the time-marching method is \((p+1)\)th order accurate and when polynomials of degree \(p\geqslant 0\) are used to represent the scalar variable, each component of the flux and the approximate trace, we observe that the approximations for the scalar variable and the flux converge with the optimal order of \(p+1\) in the \(L^{2}\)-norm. Finally, we apply element-by-element postprocessing schemes to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous interelement normal component, is shown to converge with order \(p+1\) in the \(L^{2}\)-norm. The new approximate scalar variable is shown to converge with order \(p+2\) in the \(L^{2}\)-norm. The postprocessing is performed at the element level and is thus much less expensive than the solution procedure. For the time-dependent case, the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required. Extensive numerical results are provided to demonstrate the performance of the present method.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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