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A discontinuous \(hp\) finite element method for convection-diffusion problems. (English) Zbl 0924.76051
Summary: This paper presents a new method which exhibits the best features of both finite volume and finite element techniques. Special attention is given to the issues of conservation, flexible accuracy, and stability. The method is elementwise conservative, the order of polynomial approximation can be adjusted element by element, and the stability is not based on the introduction of artifical diffusion, but on the use of a very particular finite element formulation with discontinuous basis functions. The method supports \(h\)-, \(p\)-, and \(hp\)-approximations and can be applied to any type of meshes, including non-matching grids. A priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76R99 Diffusion and convection
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[1] Allmaras, S.R., A coupled Euler/Navier-Stokes algorithm for 2-D unsteady transonic shock/boundary-layer interaction, ()
[2] Arbogast, T.; Wheeler, M.F., A characteristic-mixed finite element method for convection-dominated transport problems, SIAM J. numer. anal., 32, 404-424, (1995) · Zbl 0823.76044
[3] Arnold, D.N., An interior penalty finite element method with discontinuous elements, SIAM J. numer. anal., 19, 4, 742-760, (1982) · Zbl 0482.65060
[4] Atkins, H.L.; Shu, C.-W., Quadrature-free implementation of discontinuous Galerkin methods for hyperbolic equations, ICASE report 96-51, (1996)
[5] Aziz, A.K.; Babuška, I., The mathematical foundations of the finite element method with applications to partial differential equations, (1972), Academic Press
[6] Babuška, I.; Suri, M., The hp-version of the finite element method with quasiuniform meshes, Math. model. numer. anal., 21, 199-238, (1987) · Zbl 0623.65113
[7] I. Babu<ka, J. Tinsley Oden and C.E. Baumann, A discontinuous hp finite element method for diffusion problems: 1-D Analysis, Comput. Math. Applic., also TICAM Report 97-22, to appear.
[8] F. Bassi and R. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., submitted. · Zbl 0871.76040
[9] Bassi, F.; Rebay, R.; Savini, M.; Pedinotti, S., The discontinuous Galerkin method applied to CFD problems, ()
[10] Baumann, C.E., An hp-adaptive discontinuous finite element method for computational fluid dynamics, ()
[11] Baumann, C.E.; Oden, J. Tinsley, A discontinuous hp finite element method for the solution of the Euler equations of gas dynamics, ()
[12] Baumann, C.E.; Oden, J. Tinsley, A discontinuous hp finite element method for the solution of the Navier-Stokes equations, () · Zbl 0985.76048
[13] Bey, K.S.; Oden, J.T., A Runge-Kutta discontinuous finite element method for high speed flows, ()
[14] Bey, K.S.; Oden, J.T.; Patra, A., hp-version discontinuous Galerkin methods for hyperbolic conservation laws, Comput. methods appl. mech. engrg., 133, 3-4, 259-286, (1996) · Zbl 0894.76036
[15] Bey, K.S., An hp-adaptive discontinuous Galerkin method for hyperbolic conservation laws, () · Zbl 0860.65094
[16] Cockburn, B., An introduction to the discontinuous Galerkin method for convection-dominated problems, (1997), School of Mathematics, University of Minnesota Austin
[17] Cockburn, B.; Hou, S.; Shu, C.W., TVB Runge-Kutta local projection discontinuous Galerkin finite element for conservation laws IV: the multi-dimensional case, Math. comput., 54, 545, (1990) · Zbl 0695.65066
[18] Cockburn, B.; Hou, S.; Shu, C.W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, ICASE report 97-43, (1997)
[19] Cockburn, B.; Lin, S.Y.; Shu, C.W., TVB Runge-Kutta local projection discontinuous Galerkin finite element for conservation laws III: one-dimensional systems, J. comput. phys., 84, 90-113, (1989) · Zbl 0677.65093
[20] Cockburn, B.; Shu, C.W., TVB Runge-Kutta local projection discontinuous Galerkin finite element for conservation laws II: general framework, Math. comput., 52, 411-435, (1989) · Zbl 0662.65083
[21] B. Cockburn and C.W. Shu, The local discontinuous Galerkin method for time dependent convection-diffusion systems, SIAM J. Numer. Anal., submitted. · Zbl 0927.65118
[22] Dawson, C.N., Godunov-mixed methods for advection-diffusion equations, SIAM J. numer. anal., 30, 1315-1332, (1993) · Zbl 0791.65062
[23] Delves, L.M.; Hall, C.A., An implicit matching principle for global element calculations, J. inst. math. appl., 23, 223-234, (1979) · Zbl 0443.65087
[24] Ghia, U.; Ghia, K.N.; Shin, C.T., High-re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. comput. phys., 48, 387-411, (1982) · Zbl 0511.76031
[25] Hendry, J.A.; Delves, L.M., The global element method applied to a harmonic mixed boundary value problem, J. comput. phys., 33, 33-44, (1979) · Zbl 0438.65095
[26] Hu, C.; Shu, C.-W., A discontinuous Galerkin finite element method for the Hamilton-Jacobi equations, (1997), Division of Applied Mathematics, Brown University
[27] Johnson, C.; Pitkäranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. comput., 46, 1-26, (1986) · Zbl 0618.65105
[28] Lang, J.; Walter, A., An adaptive discontinuous finite element method for the transport equation, J. comput. phys., 117, 28-34, (1995) · Zbl 0824.65098
[29] Lesaint, P.; Raviart, P.A., On a finite element method for solving the neutron transport, (), 89-123
[30] Lesaint, P.; Raviart, P.A., Finite element collocation methods for first-order systems, Math. comput., 33, 147, 891-918, (1979) · Zbl 0417.65056
[31] I. Lomtev and G.E. Karniadakis, A discontinuous Galerkin method for the Navier-Stokes equations, Int. J. Numer. Methods Fluids, submitted. · Zbl 0951.76041
[32] Lomtev, I.; Karniadakis, G.E., Simulations of viscous supersonic flows on unstructured meshes, Aiaa-97-0754, (1997)
[33] Lomtev, I.; Quillen, C.B.; Karniadakis, G.E., Spectral/hp methods for viscous compressible flows on unstructured 2d meshes, J. comput. phys., (1998), to appear · Zbl 0929.76095
[34] Lomtev, I.; Quillen, C.W.; Karniadakis, G., Spectral/hp methods for viscous compressible flows on unstructured 2d meshes, () · Zbl 0929.76095
[35] Lowrie, R.B., Compact higher-order numerical methods for hyperbolic conservation laws, () · Zbl 0809.76077
[36] Nitsche, J., Über ein variationsprinzip zur Lösung von Dirichlet problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind, Abh. math. sem. univ. Hamburg, 36, 9-15, (1971) · Zbl 0229.65079
[37] Oden, J.T.; Carey, G.F., ()
[38] J. Tinsley Oden, I. Babuska and C.E. Baumann, A discontinuous hp finite element method for diffusion problems, J. Comput. Phys., also TICAM Report 97-21, to appear. · Zbl 0926.65109
[39] Percell, P.; Wheeler, M.F., A local residual finite element procedure for elliptic equations, SIAM J. numer. anal., 15, 4, 705-714, (1978) · Zbl 0396.65067
[40] Richter, G.R., An optimal-order error estimate for the discontinuous Galerkin method, Math. comput., 50, 75-88, (1988) · Zbl 0643.65059
[41] Warburton, T.C.; Lomtev, I.; Kirby, R.M.; Karniadakis, G.E., A discontinuous Galerkin method for the Navier-Stokes equations on hybrid grids, (1997), Center for Fluid Mechanics 97-14, Division of Applied Mathematics, Brown University
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