TY - JOUR

T1 - An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations

AU - Nguyen, N. C.

AU - Peraire, J.

AU - Cockburn, B.

PY - 2009/5/20

Y1 - 2009/5/20

N2 - We present a hybridizable discontinuous Galerkin method for the numerical solution of steady and time-dependent linear convection-diffusion equations. We devise the method as follows. First, we express the approximate scalar variable and corresponding flux within each element in terms of an approximate trace of the scalar variable along the element boundary. We then define a unique value for the approximate trace by enforcing the continuity of the normal component of the flux across the element boundary; a global equation system solely in terms of the approximate trace is thus obtained. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. If the problem is time-dependent, we discretize the time derivative by means of backward difference formulae. This results in efficient schemes capable of producing high-order accurate solutions in space and time. Indeed, when the time-marching method is (p + 1)th order accurate and when polynomials of degree p ≥ 0 are used to represent the scalar variable, the flux and the approximate trace, we observe that the approximations for the scalar variable, the flux and the trace of the scalar variable converge with the optimal order of p + 1 in the L2-norm. Finally, we introduce a simple element-by-element postprocessing scheme to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous inter-element normal component, is shown to converge with order p + 1 in the L2-norm. The new approximate scalar variable is shown to converge with order p + 2 in the L2-norm. 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. Moreover, the postprocessing procedure is less expensive than the solution procedure, since it is performed at the element level. Extensive numerical results are presented to demonstrate the convergence properties of the method.

AB - We present a hybridizable discontinuous Galerkin method for the numerical solution of steady and time-dependent linear convection-diffusion equations. We devise the method as follows. First, we express the approximate scalar variable and corresponding flux within each element in terms of an approximate trace of the scalar variable along the element boundary. We then define a unique value for the approximate trace by enforcing the continuity of the normal component of the flux across the element boundary; a global equation system solely in terms of the approximate trace is thus obtained. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. If the problem is time-dependent, we discretize the time derivative by means of backward difference formulae. This results in efficient schemes capable of producing high-order accurate solutions in space and time. Indeed, when the time-marching method is (p + 1)th order accurate and when polynomials of degree p ≥ 0 are used to represent the scalar variable, the flux and the approximate trace, we observe that the approximations for the scalar variable, the flux and the trace of the scalar variable converge with the optimal order of p + 1 in the L2-norm. Finally, we introduce a simple element-by-element postprocessing scheme to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous inter-element normal component, is shown to converge with order p + 1 in the L2-norm. The new approximate scalar variable is shown to converge with order p + 2 in the L2-norm. 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. Moreover, the postprocessing procedure is less expensive than the solution procedure, since it is performed at the element level. Extensive numerical results are presented to demonstrate the convergence properties of the method.

KW - Convection-diffusion equations

KW - Discontinuous Galerkin methods

KW - Finite element methods

KW - Hybrid/mixed methods

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U2 - 10.1016/j.jcp.2009.01.030

DO - 10.1016/j.jcp.2009.01.030

M3 - Article

AN - SCOPUS:62049083456

VL - 228

SP - 3232

EP - 3254

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 9

ER -