Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly perturbed with dominating convectio...

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Bibliographic Details
Published in:Numerische Mathematik Vol. 117; no. 2; pp. 251 - 288
Main Authors: Feistauer, Miloslav, Kučera, Václav, Najzar, Karel, Prokopová, Jaroslava
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.02.2011
Springer
Subjects:
Exact sciences and technology
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical analysis. Scientific computation
Numerical and Computational Physics
Numerical approximation
Partial differential equations
Partial differential equations, boundary value problems
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Sciences and techniques of general use
Simulation
Theoretical
65M15
65M12
65M60
Discontinuous Galerkin method
Numerical integration
Grid pattern
Error estimation
Initial value problem
Convection diffusion equation
Nonlinear problems
Boundary condition
Optimal estimation
Partial differential equation
Discretization method
Penalty method
Polynomial time
Numerical approximation
Finite element method
Interpolation
Numerical analysis
Boundary value problem
ISSN:0029-599X, 0945-3245
Online Access:Get full text
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Summary:The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “ L 2 ( L 2 )”- and “DG”-norm formed by the “ L 2 ( H 1 )”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour in time as a polynomial of degree ≤ q .
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-010-0348-x