An A Posteriori Error Estimate for the Local Discontinuous Galerkin Method Applied to Linear and Nonlinear Diffusion Problems

In this paper we present a new residual-based reliable a posteriori error estimator for the local discontinuous Galerkin approximations of linear and nonlinear diffusion problems in polygonal regions of R2. Our analysis, which applies to convex and nonconvex domains, is based on Helmholtz decompositions of the error and a suitable auxiliary polynomial function interpolating the Dirichlet datum. Several examples confirming the reliability of the estimator and providing numerical evidences for its efficiency are given. Furthermore, the associated adaptive method, which considers meshes with and without hanging nodes, is shown to be much more efficient than a uniform refinement to compute the discrete solutions. In particular, the experiments illustrate the ability of the adaptive algorithm to localize the singularities of each problem.

[1]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[2]  Gabriel N. Gatica,et al.  A Local Discontinuous Galerkin Method for Nonlinear Diffusion Problems with Mixed Boundary Conditions , 2004, SIAM J. Sci. Comput..

[3]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[4]  Carsten Carstensen,et al.  An a posteriori error estimate for a first-kind integral equation , 1997, Math. Comput..

[5]  Salim Meddahi,et al.  A dual-dual mixed formulation for nonlinear exterior transmission problems , 2001, Math. Comput..

[6]  Norbert Heuer,et al.  An expanded mixed finite element approach via a dual-dual formulation and the minimum residual method , 2001 .

[7]  Carsten Carstensen,et al.  A posteriori error estimates for nonconforming finite element methods , 2002, Numerische Mathematik.

[8]  Ilaria Perugia,et al.  An hp-Analysis of the Local Discontinuous Galerkin Method for Diffusion Problems , 2002, J. Sci. Comput..

[9]  P. Hansbo,et al.  A finite element method for domain decomposition with non-matching grids , 2003 .

[10]  D. Arnold.  An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

[11]  Jindřich Nečas,et al.  Introduction to the Theory of Nonlinear Elliptic Equations , 1986 .

[12]  Ilaria Perugia,et al.  On the Coupling of Local Discontinuous Galerkin and Conforming Finite Element Methods , 2001, J. Sci. Comput..

[13]  Peter Hansbo,et al.  Energy norm a posteriori error estimation for discontinuous Galerkin methods , 2003 .

[14]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[15]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[16]  Clint Dawson,et al.  Some Extensions Of The Local Discontinuous Galerkin Method For Convection-Diffusion Equations In Mul , 1999 .

[17]  Ilaria Perugia,et al.  An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems , 2000, SIAM J. Numer. Anal..

[18]  Ilaria Perugia,et al.  Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids , 2001, SIAM J. Numer. Anal..