zbMATH — the first resource for mathematics

Time discretization of parabolic problems by the discontinuous Galerkin method. (English) Zbl 0589.65070
The authors analyze the discontinuous Galerkin method for the time discretization of parabolic type problems \(y_ t+Ay=f\) for \(t\geq 0\), \(y(0)=y_ 0\), where y is a function of t with values in a Hilbert space H, A is a self-adjoint positive definite linear operator on H (independent of t), and \(y_ 0\) and \(f=f(t)\) are given data. Error estimates are derived at the nodal points as well as uniformly in time for smooth and non-smooth initial data. These estimates are then combined with known estimates for semi-discrete in space Galerkin approximations of parabolic problems to yield error estimates for complete discretizations of such problems.
Reviewer: W.Ames

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
Full Text: DOI EuDML
[1] G.A. BAKER, J. H. BRAMBLE and V. THOMÉE, Single step Galerkin approximations for parabolic problems. Math. comp. 31, 818-847 (1977). Zbl0378.65061 MR448947 · Zbl 0378.65061 · doi:10.2307/2006116
[2] M. C. DELFOUR, W.W. HAGER and F. TROCHU, Discontinuous Galerkin methods for ordinary differential equations. Math. Comp. 36, 455-473 (1981). Zbl0469.65053 MR606506 · Zbl 0469.65053 · doi:10.2307/2007652
[3] P. JAMET, Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain. SIAM J. Numer. Anal. 15, 912-928 (1978). Zbl0434.65091 MR507554 · Zbl 0434.65091 · doi:10.1137/0715059
[4] C. JOHNSON, On error estimates for numerical methods for stiff o.d.e’s. Preprint, Department of Mathematics, University of Michigan, 1984.
[5] M. LUSKIN and R. RANNACHER, On the smoothing property of the Galerkin method for parabolic equations SIAM J. Numer. Anal. 19, 93-113 (1981). Zbl0483.65064 MR646596 · Zbl 0483.65064 · doi:10.1137/0719003
[6] V. THOMÉE, Galerkin Methods for Parabolic Problems, Springer Lecture Notes in Mathematics, No. 1054, 1984. Zbl0528.65052 MR744045 · Zbl 0528.65052 · doi:10.1007/BFb0071790
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.