Some Error Estimates for the Discretization of Parabolic Equations on General Multidimensional Nonconforming Spatial Meshes

This work is devoted to error estimates for the discretization of parabolic equations on general nonconforming spatial meshes in several space dimensions. These meshes have been recently used to approximate stationary anisotropic heterogeneous diffusion equations and nonlinear equations. We present an implicit time discretization scheme based on an orthogonal projection of the exact initial value. We prove that, when the discrete flux is calculated using a stabilized discrete gradient, the convergence order is \(h_{\mathcal{D}}+k\), where \(h_{\mathcal{D}}\) (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid for discrete norms \({\mathbb{L}}^\infty(0,T;H^1_0(\Omega))\) and \({\mathcal W}^{1,\infty}(0,T;L^2(\Omega))\) under the regularity assumption \(u\in {\mathcal{C}}^2([0,T];{\mathcal{C}}^2(\overline{\Omega}))\) for the exact solution u. These error estimates are useful because they allow to obtain approximations to the exact solution and its first derivatives of order \(h_{\mathcal{D}}+k\).

Authors and Affiliations

1. Department of Mathematics, University of Annaba, Algeria

   Abadallah Bradji

2. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117, Berlin, Germany

   Jürgen Fuhrmann

Editors and Affiliations

1. Institute of Computer and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev 25 A, 1113, Sofia, Bulgaria

   Ivan Dimov

2. Faculty of Mathematics and Informatics, Department Numerical Methods and Algorithms, University of Sofia "St. Kliment Ohridski",, blvd. James Bourchier 5, 1164, Sofia, Bulgaria

   Stefka Dimova

3. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. Bonchev St.,Bl.8, 1113, Sofia, Bulgaria

   Natalia Kolkovska

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