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Volumn 9, Issue 2, 2001, Pages 99-122

On the relationship of various discontinuous finite element methods for second-order elliptic equations

Author keywords

convergence; Discontinuous finite element methods; elliptic equations; equivalence; parabolic equations; stability

Indexed keywords

APPROXIMATION THEORY; ASYMPTOTIC STABILITY; BOUNDARY CONDITIONS; CONVERGENCE OF NUMERICAL METHODS; ERROR ANALYSIS; GALERKIN METHODS; LAGRANGE MULTIPLIERS; PARTIAL DIFFERENTIAL EQUATIONS; TENSORS;

EID: 0034930339     PISSN: 15702820     EISSN: None     Source Type: Journal    
DOI: 10.1515/JNMA.2001.99     Document Type: Article
Times cited : (16)

References (52)
  • 1
    • 0003212031 scopus 로고
    • Lectures on Elliptic Boundary Value Problems
    • S. Agmon, Lectures on Elliptic Boundary Value Problems. Van Nostrand, Princeton, NJ, 1965.
    • (1965) Van Nostrand, Princeton, NJ
    • Agmon, S.1
  • 2
    • 85025286282 scopus 로고    scopus 로고
    • The local discontinuous Galerkin method for contaminant transport, Tech
    • V. Aizinger, C. Dawson, B. Cockburn, and P. Castillo, The local discontinuous Galerkin method for contaminant transport, Tech. Report No. 99-39. TICAM, University of Texas at Austin, 1999.
    • (1999) Report No. , pp. 39-99
    • Aizinger, V.1    Dawson, C.2    Cockburn, B.3    Castillo, P.4
  • 3
    • 84968500702 scopus 로고
    • Chen, On the implementation of mixed methods as nonconforming methods for second order elliptic problems
    • T. J. Arbogast and Zh. Chen, On the implementation of mixed methods as nonconforming methods for second order elliptic problems. Math. Comp. (1995) 64, 943-972.
    • (1995) Math. Comp. , vol.64 , pp. 943-972
    • Arbogast, T.J.1
  • 4
    • 0001163226 scopus 로고
    • An interior penalty finite element method with discontinuous elements
    • D. N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. (1982) 19, 742-760.
    • (1982) SIAM J. Numer. Anal. , vol.19 , pp. 742-760
    • Arnold, D.N.1
  • 5
    • 0001804939 scopus 로고
    • Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates
    • D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modl. Math. Anal. Numr. (1985) 19, 7-32.
    • (1985) RAIRO Modl. Math. Anal. Numr. , vol.19 , pp. 7-32
    • Arnold, D.N.1    Brezzi, F.2
  • 6
    • 0003941155 scopus 로고    scopus 로고
    • Quadrature-free implementation of discontinuous Galerkin methods for hyperbolic equations
    • H. L. Atkins and C.-W. Shu, Quadrature-free implementation of discontinuous Galerkin methods for hyperbolic equations. ICASE Report No. 96-51, 1996.
    • (1996) ICASE Report
    • Atkins, H.L.1    Shu, C.-W.2
  • 7
    • 84966227273 scopus 로고
    • The finite element method with penalty
    • I. Babuska, The finite element method with penalty. Math. Comp. (1973) 27, 221 - 228.
    • (1973) Math. Comp. , vol.27 , pp. - 228
    • Babuska, I.1
  • 8
    • 0017494499 scopus 로고
    • Mixed-hybrid finite element approximations of second- order elliptic boundary-value problems
    • I.Babuska, J.T.Oden, and J.K.Lee, Mixed-hybrid finite element approximations of second- order elliptic boundary-value problems. Comp. Meth. Appl. Mech. Engrg. (1977) 11, 176-189.
    • (1977) Comp. Meth. Appl. Mech. Engrg. , vol.11 , pp. 176-189
    • Babuska, I.1    Oden, J.T.2    Lee, J.K.3
  • 9
    • 0001537143 scopus 로고
    • Nonconforming elements in the finite element method with penalty
    • I. Babuska and M. Zlmal, Nonconforming elements in the finite element method with penalty. SIAMJ. Numer. Anal. (1973) 10, 863-875.
    • (1973) SIAMJ. Numer. Anal. , vol.10 , pp. 863-875
    • Babuska, I.1    Zlmal, M.2
  • 10
    • 84904152686 scopus 로고
    • Finite element methods for elliptic equations using nonconforming elements
    • G. A. Baker, Finite element methods for elliptic equations using nonconforming elements. Math. Comp. (1977) 31, 45-59.
    • (1977) Math. Comp. , vol.31 , pp. 45-59
    • Baker, G.A.1
  • 11
    • 0001416032 scopus 로고    scopus 로고
    • A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations
    • F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comp. Phys. (1997) 131, 267 - 279.
    • (1997) J. Comp. Phys. , vol.131 , pp. - 279
    • Bassi, F.1    Rebay, S.2
  • 12
    • 0031537299 scopus 로고    scopus 로고
    • High-order accurate discontinuous finite element solution of the 2D Eulerequations
    • F. Bassi and S. Rebay, High-order accurate discontinuous finite element solution of the 2D Eulerequations. J. Comp. Phys. (1997) 138, 251-285.
    • (1997) J. Comp. Phys. , vol.138 , pp. 251-285
    • Bassi, F.1    Rebay, S.2
  • 13
    • 0003347887 scopus 로고
    • A Runge-Kutta discontinuous finite element method for high speed flows
    • AIAA
    • K. S. Bey and J. T. Oden, A Runge-Kutta discontinuous finite element method for high speed flows.In: AIAA 10th Computational Fluid Dynamics Conference, June 1991.
    • (1991)
    • Bey, K.S.1    Oden, J.T.2
  • 14
    • 0028407037 scopus 로고
    • Parallel, adaptive finite element methods for conservation laws
    • R. Biswas, K. D. Devine, and J. Flaherty, Parallel, adaptive finite element methods for conservation laws. Appl. Numer. Math. (1994) 14, 255-283.
    • (1994) Appl. Numer. Math. , vol.14 , pp. 255-283
    • Biswas, R.1    Devine, K.D.2    Flaherty, J.3
  • 15
    • 0003199785 scopus 로고
    • Mixed finite elements for second order elliptic problems in three variables
    • F. Brezzi, J. Douglas, Jr., R. Duran, and M. Fortin, Mixed finite elements for second order elliptic problems in three variables. Numer. Math. (1987) 51, 237 - 250.
    • (1987) Numer. Math. , vol.51 , pp. - 250
    • Brezzi, F.1    Douglas, J.2
  • 16
    • 0001165889 scopus 로고
    • Efficient rectangular mixed finite elements in two and three space variables
    • F. Brezzi, J. Douglas, Jr., M. Fortin, and L. D. Marini, Efficient rectangular mixed finite elements in two and three space variables. RAIRO Model. Math. Anal. Numr. (1987) 21, 581 - 604.
    • (1987) RAIRO Model. Math. Anal. Numr. , vol.21 , pp. - 604
    • Brezzi, F.1    Douglas, J.2
  • 17
    • 34250118577 scopus 로고
    • Two families of mixed finite elements for second orderelliptic problems
    • F.Brezzi, J.Douglas, Jr., and L.D.Marini, Two families of mixed finite elements for second orderelliptic problems. Numer. Math. (1985) 47, 217-235.
    • (1985) Numer. Math. , vol.47 , pp. 217-235
    • Douglas, J.1
  • 20
    • 0021587596 scopus 로고
    • Discontinuous upwinding and mixed finite elements for two-phase flows in reservoir simulation
    • G. Chavent and J. Jaffre, Discontinuous upwinding and mixed finite elements for two-phase flows in reservoir simulation. Comp. Meth. Appl. Mech. Engrg. (1984) 47, 93 -118.
    • (1984) Comp. Meth. Appl. Mech. Engrg. , vol.47 , pp. -118
    • Chavent, G.1    Jaffre, J.2
  • 21
    • 0002444350 scopus 로고
    • Analysis of mixed methods using conforming and nonconfirming finite element methods
    • Zh. Chen, Analysis of mixed methods using conforming and nonconfirming finite element methods. RAIRO Model. Math. Anal. Numer. (1993) 27, 9-34.
    • (1993) RAIRO Model. Math. Anal. Numer. , vol.27 , pp. 9-34
    • Chen, Z.1
  • 22
    • 0001167127 scopus 로고
    • BDM mixed methods for a nonlinear elliptic problem
    • Zh. Chen, BDM mixed methods for a nonlinear elliptic problem. J. Comp. Appl. Math. (1994) 53, 207-223.
    • (1994) J. Comp. Appl. Math. , vol.53 , pp. 207-223
    • Chen, Z.1
  • 23
    • 0029697285 scopus 로고    scopus 로고
    • Equivalence between and multigrid algorithms for nonconforming and mixed methods for second order elliptic problems
    • Zh. Chen, Equivalence between and multigrid algorithms for nonconforming and mixed methods for second order elliptic problems. East-West J. Numer. Math. (1996) 4, No. 1, 1 -33.
    • (1996) East-West J. Numer. Math. , pp. -33
    • Chen, Z.1
  • 24
    • 0004577851 scopus 로고    scopus 로고
    • Characteristics-discontinuous finite element methods in mixed form for advection- dominated diffusion problems
    • Zh. Chen, Characteristics-discontinuous finite element methods in mixed form for advection- dominated diffusion problems. SMUMath. Report No. 2000-04, Southern Methodist University, Dallas, 2000.
    • (2000) SMUMath. Report No. 2000-04, Southern Methodist University, Dallas
    • Chen, Z.1
  • 25
    • 0000449846 scopus 로고
    • Quantum hydrodynamic simulation of hysteresis inthe resonant tunneling diode
    • Zh.Chen, B.Cockburn, C.Gardner, and J.W.Jerome, Quantum hydrodynamic simulation of hysteresis inthe resonant tunneling diode. J. Comp. Phys. (1995) 117, 274-280.
    • (1995) J. Comp. Phys. , vol.117 , pp. 274-280
    • Cockburn, B.1
  • 26
    • 0029482835 scopus 로고
    • Mixed-RKDG finite element methods for the 2-D hydrodynamic model for semiconductor device simulation
    • Zh. Chen, B. Cockburn, J. W. Jerome, and C.-W. Shu, Mixed-RKDG finite element methods for the 2-D hydrodynamic model for semiconductor device simulation. VLSI Designs (1995) 3, 145-158.
    • (1995) VLSI Designs , vol.3 , pp. 145-158
    • Chen, Z.1    Cockburn, B.2    Jerome, J.W.3    Shu, C.-W.4
  • 27
    • 51249179041 scopus 로고
    • Prismatic mixed finite elements for second order elliptic problems
    • Zh. Chen and J. Douglas, Jr., Prismatic mixed finite elements for second order elliptic problems. Calcolo (1989) 26, 135-148.
    • (1989) Calcolo , vol.26 , pp. 135-148
    • Chen, Z.1    Douglas, J.2
  • 28
    • 0004632908 scopus 로고    scopus 로고
    • Stability and convergence of a finite element method for reactive transport in ground water
    • Zh. Chen and R. E. Ewing, Stability and convergence of a finite element method for reactive transport in ground water. SIAM J. Numer. Anal. (1997) 34, 881 -904.
    • (1997) SIAM J. Numer. Anal. , vol.34 , pp. -904
    • Chen, Z.1    Ewing, R.E.2
  • 30
    • 84966246413 scopus 로고
    • TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: general framework
    • B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: general framework. Math. Comp. (1989) 52, 411-435.
    • (1989) Math. Comp. , vol.52 , pp. 411-435
    • Cockburn, B.1    Shu, C.-W.2
  • 31
    • 0344393366 scopus 로고    scopus 로고
    • The local discontinuous Galerkin method for time-dependent convection diffusion systems
    • B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection diffusion systems. SIAM J. Numer. Anal. (1998) 35, 2440-2463.
    • (1998) SIAM J. Numer. Anal. , vol.35 , pp. 2440-2463
    • Cockburn, B.1    Shu, C.-W.2
  • 32
    • 77954673424 scopus 로고
    • An implicit matching principle for global element calculations.J
    • L. M. Delves and C. A. Hall, An implicit matching principle for global element calculations.J. Inst. Math. Appl. (1979) 23, 223-234.
    • (1979) Inst. Math. Appl. , vol.23 , pp. 223-234
    • Delves, L.M.1    Hall, C.A.2
  • 33
    • 0001456343 scopus 로고
    • Interior penalty procedures for elliptic and parabolic Galerkin methods
    • Springer-Verlag
    • J. Douglas, Jr. and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods. Lecture Notes in Physics, Vol. 58, Springer-Verlag, Berlin, 1976, pp. 207-216.
    • (1976) Lecture Notes in Physics , vol.58 , pp. 207-216
    • Douglas, J.1
  • 34
    • 0002329650 scopus 로고
    • The approximation of the pressure by a mixed method inthe simulation of miscible displacement
    • J.Douglas, Jr., R.E.Ewing, and M.Wheeler, The approximation of the pressure by a mixed method inthe simulation of miscible displacement. RAIRO Anal. Numer. (1983) 17, 17-33.
    • (1983) RAIRO Anal. Numer. , vol.17 , pp. 17-33
  • 35
    • 0026106415 scopus 로고
    • Adaptive finite element methods for parabolic problems I: A linear model problem
    • K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems I: A linear model problem. SIAM J. Numer. Anal. (1991) 28, 43-77.
    • (1991) SIAM J. Numer. Anal. , vol.28 , pp. 43-77
    • Eriksson, K.1    Johnson, C.2
  • 36
    • 0001188018 scopus 로고
    • Time discretization of parabolic problems by the discontinuous Galerkin method
    • K. Eriksson, C. Johnson, and V. Thomee, Time discretization of parabolic problems by the discontinuous Galerkin method. RAIRO Model. Math. Anal. Numer (1985) 19, 611-643.
    • (1985) RAIRO Model. Math. Anal. Numer , vol.19 , pp. 611-643
    • Eriksson, K.1    Johnson, C.2    Thomee, V.3
  • 38
    • 0000107217 scopus 로고    scopus 로고
    • Explicit finite element methods for symmetric hyperbolic equations
    • R.S.Falk and G.R.Richter, Explicit finite element methods for symmetric hyperbolic equations. Math. Comp. (1999) 70, 935-952.
    • (1999) Math. Comp. , vol.70 , pp. 935-952
  • 39
    • 84966238599 scopus 로고
    • An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation
    • C. Johnson and J. Pitkaranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp. (1986) 46, 1 -26.
    • (1986) Math. Comp. , vol.46 , pp. -26
    • Johnson, C.1    Pitkaranta, J.2
  • 41
    • 0004610152 scopus 로고    scopus 로고
    • An optimal-order error estimate of the discontinuous Galerkin method
    • Q. Lin, N. Yan, and A.-H. Zou, An optimal-order error estimate of the discontinuous Galerkin method. J. Engin. Math. (1996) 13, 101 -105.
    • (1996) J. Engin. Math. , vol.13 , pp. -105
    • Lin, Q.1    Yan, N.2    Zou, A.-H.3
  • 42
    • 0003828734 scopus 로고    scopus 로고
    • Compact higher-order numerical methods for hyperbolic conservation laws
    • University of Michigan
    • R. B. Lowrie, Compact higher-order numerical methods for hyperbolic conservation laws. Ph.D. Dissertation, University of Michigan, 1996.
    • (1996) Ph.D. Dissertation
    • Lowrie, R.B.1
  • 43
    • 0022077278 scopus 로고
    • An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method
    • L. D. Marini, An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. SIAM J. Numer. Anal. (1985) 22, 493-496.
    • (1985) SIAM J. Numer. Anal. , vol.22 , pp. 493-496
    • Marini, L.D.1
  • 44
    • 34250256485 scopus 로고
    • Mixed finite elements in IR3
    • J. Nedelec, Mixed finite elements in IR3. Numer. Math. (1980) 35, 315 -341.
    • (1980) Numer. Math. , vol.35 , pp. -341
    • Nedelec, J.1
  • 45
    • 84951671794 scopus 로고
    • Uber ein variationsprinzip zur losung von Dirichlet problem bei verwendung von teilraumen, die keinen randbedingungen unterworfen sind
    • J. A. Nitsche, Uber ein variationsprinzip zur losung von Dirichlet problem bei verwendung von teilraumen, die keinen randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg (1971) 36, 9-15.
    • (1971) Abh. Math. Sem. Univ. Hamburg , vol.36 , pp. 9-15
    • Nitsche, J.A.1
  • 46
    • 0032209437 scopus 로고    scopus 로고
    • A discontinuous hp finite element method for diffusion problems
    • J. T. Oden, I. Babuska, and C. E. Baumann, A discontinuous hp finite element method for diffusion problems. J. Comp. Phys. (1998) 146, 491 -519.
    • (1998) J. Comp. Phys. , vol.146 , pp. -519
    • Oden, J.T.1    Babuska, I.2    Baumann, C.E.3
  • 47
    • 0001175293 scopus 로고
    • A mixed finite element method for second order elliptic problems
    • Springer
    • R. Raviart, and J. Thomas, A mixed finite element method for second order elliptic problems. Lecture Notes in Mathematics, Vol. 606, Springer, Berlin, 1977, pp. 292-315.
    • (1977) Lecture Notes in Mathematics , vol.606 , pp. 292-315
    • Raviart, R.1    Thomas, J.2
  • 49
    • 84966227504 scopus 로고
    • An optimal-order error estimate for the discontinuous Galerkin method
    • G. R. Richter, An optimal-order error estimate for the discontinuous Galerkin method. Math. Comp. (1988) 50, 75-88.
    • (1988) Math. Comp. , vol.50 , pp. 75-88
    • Richter, G.R.1
  • 50
    • 0004029319 scopus 로고    scopus 로고
    • Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems
    • B. Riviere and M. F. Wheeler, Part I. Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Tech. Report No. 99-09, TICAM, University of Texas at Austin, 1999.
    • (1999) Tech. Report No. 99-09, TICAM, University of Texas at Austin
    • Riviere, B.1    Wheeler, M.F.2
  • 52
    • 0000802762 scopus 로고
    • An elliptic collocation-finite element method with interior penalties
    • M. F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. (1978) 15, 152-161.
    • (1978) SIAM J. Numer. Anal. , vol.15 , pp. 152-161
    • Wheeler, M.F.1


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