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A discontinuous $$hp$$ finite element method for diffusion problems: 1-D analysis. (English) Zbl 0940.65076
The mathematical background for a new type of discontinuous Galerkin method is reported, including stability analysis and error estimates, and is illustrated by the numerical analysis of a steady state 1-D diffusion problem. The shape functions are discontinuous at the element boundaries. The main features of the method are: auxiliary variables are not needed; robustness; the method yields elementwise conservative approximations; resulting matrices are positive-definite, well conditioned and convenient for the treatment of transient problems; adaptive error control; acceptable costs.

##### MSC:
 65L10 Numerical solution of boundary value problems involving ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 65L70 Error bounds for numerical methods for ordinary differential equations
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##### References:
 [1] 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 [2] Percell, P.; Wheeler, M.F., A local residual finite element procedure for elliptic equations, SIAM J. numer. anal., 15, 4, 705-714, (August 1978) [3] Wheeler, M.F., An elliptic collocation-finite element method with interior penalties, SIAM J. numer. anal., 15, 4, 152-161, (1978) · Zbl 0384.65058 [4] Arnold, D.N., An interior penalty finite element method with discontinuous elements, SIAM J. numer. anal., 19, 4, 742-760, (August 1982) [5] 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 [6] Hendry, J.A.; Delves, L.M., The global element method applied to a harmonic mixed boundary value problem, J. comp. phys., 33, 33-44, (1979) · Zbl 0438.65095 [7] Baumann, C.E., An hp-adaptive discontinuous finite element method for computational fluid dynamics, () [8] Oden, J.T.; Babuška, I.; Baumann, C.E., A discontinuous hp finite element method for diffusion problems, TICAM report 97-21, (1997) [9] Dawson, C.N., Godunov-mixed methods for advection-diffusion equations, SIAM J. numer. anal., 30, 1315-1332, (1993) · Zbl 0791.65062 [10] 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 [11] Bassi, F.; Rebay, R.; Savini, M.; Pedinotti, S., The discontinuous Galerkin method applied to CFD problems, () [12] Bassi, F.; Rebay, R., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. comp. physics, 131, 2, 267-279, (1997) · Zbl 0871.76040 [13] Lomtev, I.; Quillen, C.W.; Karniadakis, G., Spectral/hp methods for viscous compressible flows on meshes, () · Zbl 0929.76095 [14] Lomtev, I.; Karniadakis, G.E., Simulations of viscous supersonic flows on unstructured meshes, (1997), AIAA-97-0754 [15] Lomtev, I.; Quillen, C.B.; Karniadakis, G.E., Spectral/hp methods for viscous compressible flows on unstructured 2d meshes, J. comp. phys., 144, 2, 325-357, (1998) · Zbl 0929.76095 [16] Lomtev, I.; Karniadakis, G.E., A discontinuous Galerkin method for the Navier-Stokes equations, Int. J. num. meth. fluids, (1997), (submitted) [17] Warburton, T.C.; Lomtev, I.; Kirby, R.M.; Karniadakis, G.E., A discontinuous Galerkin method for the Navier-Stokes equations on hybrid grids, () · Zbl 0992.76056 [18] Cockburn, B.; Shu, C.W., The local discontinuous Galerkin method for time dependent convection-diffusion systems, SIAM J. numer. anal., (1997), (submitted) [19] Cockburn, B., An introduction to the discontinuous Galerkin method for convection-dominated problems, (1997), School of Mathematics, University of Minnesota [20] Aziz, A.K.; Babuška, I., The mathematical foundations of the finite element method with applications to partial differential equations, (1972), Academic Press [21] Oden, J.T.; Carey, G.F., Texas finite elements series vol. IV—mathematical aspects, (1983), Prentice-Hall [22] Babuska, I.; Suri, M., The hp-version of the finite element method with quasiuniform meshes, Mathematical modeling and numerical analysis, 21, 199-238, (1987) · Zbl 0623.65113 [23] Babuška, I.; Oden, J.T.; Baumann, C.E., A discontinuous hp finite element method for diffusion problems: 1-D analysis, TICAM report 97-22, (1997)
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