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Discrete maximum principle for interior penalty discontinuous Galerkin methods. (English) Zbl 1269.65124

This is probably the first paper devoted to the discrete maximum principle (DMP) for the discontinuous Galerkin method. It overviews the definitions and results for the continuous and discrete maximum principles for elliptic operators and their finite element discretizations. Then the authors consider the one-dimensional reaction-diffusion operator \(K u = -(p u')' + k^2 u\) with constants \(p>0\) and \(k\) and discretize it by the interior penalty discontinuous Galerkin method. The main result provides conditions for the validity of the DMP for the resulting discrete operator. These conditions are formulated in therms of the parameters \(p\), \(k\), penalty parameter and the mesh size. Finally, a numerical example is presented.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35B50 Maximum principles in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
34B05 Linear boundary value problems for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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