An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems

An efficient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator. An adaptive algorithm with variable time-step sizes and space meshes is proposed and studied which, at each time step, delays the mesh coarsening until the final iteration of the adaptive procedure, allowing only mesh and time-step size refinements before. It is proved that at each time step the adaptive algorithm is able to reduce the error indicators (and thus the error) below any given tolerance within a finite number of iteration steps. The key ingredient in the analysis is a new coarsening strategy. Numerical results are presented to show the competitive behavior of the proposed adaptive algorithm.

[1]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems IV: nonlinear problems , 1995 .

[2]  Ricardo H. Nochetto,et al.  Error Control and Andaptivity for a Phase Relaxation Model , 2000 .

[3]  Ricardo H. Nochetto,et al.  A characteristic Galerkin method with adaptive error control for the continuous casting problem , 2000 .

[4]  I. Babuška,et al.  The finite element method for parabolic equations , 1982 .

[5]  Kunibert G. Siebert,et al.  ALBERT: An adaptive hierarchical nite element toolbox , 2000 .

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

[7]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

[8]  Ricardo H. Nochetto,et al.  Data Oscillation and Convergence of Adaptive FEM , 2000, SIAM J. Numer. Anal..

[9]  Eberhard Bänsch,et al.  Local mesh refinement in 2 and 3 dimensions , 1991, IMPACT Comput. Sci. Eng..

[10]  Zhiming Chen,et al.  Adaptive Galerkin Methods with Error Control for a Dynamical Ginzburg-Landau Model in Superconductivity , 2000, SIAM J. Numer. Anal..

[11]  M. Picasso.  Adaptive finite elements for a linear parabolic problem , 1998 .

[12]  Zhiming Chen,et al.  On the Efficiency of Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients , 2002, SIAM J. Sci. Comput..

[13]  Rüdiger Verfürth,et al.  A posteriori error estimates for nonlinear problems. Lr(0, T; Lrho(Omega))-error estimates for finite element discretizations of parabolic equations , 1998, Math. Comput..

[14]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

[15]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems. I.: a linear model problem , 1991 .

[16]  Ivo Babuška,et al.  The finite element method for parabolic equations , 1982 .

[17]  Rüdiger Verfürth,et al.  A posteriori error estimates for nonlinear problems , 1994 .

[18]  Peter K. Moore,et al.  A posteriori error estimation with finite element semi- and fully discrete methods for nonlinear parabolic equations in one space dimension , 1994 .

[19]  Ricardo H. Nochetto,et al.  A posteriori error estimation and adaptivity for degenerate parabolic problems , 2000, Math. Comput..

[20]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[21]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .