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Ayuso de Dios, Blanca; Brezzi, Franco; Havle, Oto; Marini, Luisa Donatella

\(L^2\)-estimates for the DG IIPG-0 scheme. (English) Zbl 1256.65095

Numer. Methods Partial Differ. Equations 28, No. 5, 1440-1465 (2012).
The known error estimate of the approximate solution \( u_{h}\) of the problem \[ - \Delta u= f\quad\text{ in}\,\, \Omega\subset \mathbb{R}^{d}, ~u\in H^{2}(\Omega),~ f\in L^{2}(\Omega), \] obtained by the discontinuous Galerkin method is established for the same problem studied in the space \(L^{2}(\Omega)\). The space of the finite elements is constructed using the decomposition of \(\Omega\) into regular triangles \(T\) (or tetrahedrons if \(d=3\)).
Reviewer: Ivan Secrieru (Chişinău)

Cited in 7 Documents

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs

Keywords:

Dirichlet problem; finite element method; error estimate; Poisson equation; discontinuous Galerkin method
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References:

[1] Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J Numer Anal 15 pp 152– (1978) · Zbl 0384.65058 · doi:10.1137/0715010
[2] Arnold, An interior penalty finite element method with discontinuous elements, SIAM J Numer Anal 19 pp 742– (1982) · Zbl 0482.65060 · doi:10.1137/0719052
[3] Arnold, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J Numer Anal 39 pp 1749– (2001) · Zbl 1008.65080 · doi:10.1137/S0036142901384162
[4] Larson, Analysis of a family of discontinuous Galerkin methods for elliptic problems: the one dimensional case, Numer Math 99 pp 113– (2004) · Zbl 1064.65082 · doi:10.1007/s00211-004-0528-7
[5] Chen, Superconvergence properties of discontinuous Galerkin methods for two-point boundary value problems, Int J Numer Anal Model 3 pp 163– (2006) · Zbl 1110.65074
[6] Guzmán, Sub-optimal convergence of non-symmetric discontinuous Galerkin methods for odd polynomial approximations, J Sci Comput 40 pp 273– (2009) · Zbl 1203.65255 · doi:10.1007/s10915-008-9255-z
[7] Dolej&0161;í, The L2 -optimality of the IIPG method for odd degrees of polynomial approximation in 1D, J Sci Comput 42 pp 122– (2010) · Zbl 1203.65251 · doi:10.1007/s10915-009-9319-8
[8] Burman, Low order discontinuous Galerkin methods for second order elliptic problems, SIAM J Numer Anal 47 pp 508– (2008) · Zbl 1190.65170 · doi:10.1137/070685105
[9] Wang, An optimal-order L2 -error estimate for nonsymmetric discontinuous Galerkin methods for a parabolic equation in multiple space dimensions, Comput Methods Appl Mech Eng 198 pp 2190– (2009) · Zbl 1227.65085 · doi:10.1016/j.cma.2009.02.006
[10] Ayuso de Dios, Uniformly convergent iterative methods for discontinuous Galerkin discretizations, J Sci Comput 40 pp 4– (2009) · Zbl 1203.65242 · doi:10.1007/s10915-009-9293-1
[11] Adams, Pure and Applied Mathematics 65 (1975)
[12] Brenner, Poincaré-Friedrichs inequalities for piecewise H1 functions, SIAM J Numer Anal 41 pp 306– (2003) · Zbl 1045.65100 · doi:10.1137/S0036142902401311
[13] S. Agmon Lectures on elliptic boundary value problems B. Frank Jones Jr. G. W. Batten Jr. Van Nostrand Mathematical Studies, No. 2 D. Van Nostrand Co., Inc. Princeton, N. J.-Toronto-London 1965
[14] Sun, Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media, SIAM J Numer Anal 43 pp 195– (2005) · Zbl 1086.76043 · doi:10.1137/S003614290241708X
[15] Brenner, A weakly over-penalized non-symmetric interior penalty method, J Numer Anal Ind Appl Math 2 pp 35– (2007) · Zbl 1145.65095
[16] Brenner, A weakly over-penalized symmetric interior penalty method, Electron Trans Numer Anal 30 pp 107– (2008) · Zbl 1171.65077
[17] Brezzi, Stabilization mechanisms in discontinuous Galerkin finite element methods, Comput Methods Appl Mech Eng 195 pp 3293– (2006) · Zbl 1125.65102 · doi:10.1016/j.cma.2005.06.015
[18] Brezzi, Mixed discontinuous Galerkin methods for Darcy flow, J Sci Comput 22/23 pp 119– (2005) · Zbl 1103.76031 · doi:10.1007/s10915-004-4150-8
[19] Lin, Linear finite elements with high accuracy, J Comput Math 3 pp 115– (1985) · Zbl 0577.65094
[20] Bank, Asymptotically exact a posteriori error estimators. I. Grids with superconvergence, SIAM J Numer Anal 41 pp 2294– (2003) · Zbl 1058.65116 · doi:10.1137/S003614290139874X
[21] Ciarlet, Handbook of Numerical Analysis II 2, in: Handbook of numerical analysis pp 17– (1991) · doi:10.1016/S1570-8659(05)80039-0
[22] Rivière, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. I, Comput Geosci 3 pp 337– (1999) · Zbl 0951.65108 · doi:10.1023/A:1011591328604
[23] Rivière, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J Numer Anal 39 pp 902– (2001) · Zbl 1010.65045 · doi:10.1137/S003614290037174X
[24] Rudin, Functional analysis (1973)
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