Analysis of a robust finite element approximation for a parabolic equation with rough boundary data

The approximation of parabolic equations with nonhomogeneous Dirichlet boundary data by a numerical method that consists of finite elements for the space discretization and the backward Euler time discretization is studied. The boundary values are assumed in a least squares sense. It is shown that this method achieves an optimal rate of convergence for rough (only L2) boundary data and for smooth data as well. The results of numerical computations which confirm the robust theoretical error estimates are also presented.

[1]  Vidar Thomée,et al.  An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem , 1990 .

[2]  Paul L. Butzer,et al.  Semi-groups of operators and approximation , 1967 .

[3]  I. Lasiecka,et al.  Galerkin approximations of abstract parabolic boundary value problems with rough boundary data— L p theory , 1986 .

[4]  Greg Knowles,et al.  Finite Element Approximation of Parabolic Time Optimal Control Problems , 1982 .

[5]  R. Winther.  Error estimates for a galerkin approximation of a parabolic control problem , 1978 .

[6]  Vidar Thomée,et al.  Discrete time Galerkin methods for a parabolic boundary value problem , 1974 .

[7]  Max Gunzburger,et al.  On finite element approximations of problems having inhomogeneous essential boundary conditions , 1983 .

[8]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[9]  Donald A. French,et al.  Approximation of an elliptic control problem by the finite element method , 1991 .

[10]  I. Lasiecka.  Unified theory for abstract parabolic boundary problems—a semigroup approach , 1980 .

[11]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[12]  P. Grisvard,et al.  BEHAVIOR OF THE SOLUTIONS OF AN ELLIPTIC BOUNDARY VALUE PROBLEM IN A POLYGONAL OR POLYHEDRAL DOMAIN , 1976 .

[13]  Randolph E. Bank,et al.  An optimal order process for solving finite element equations , 1981 .

[14]  J. Pasciak,et al.  The Construction of Preconditioners for Elliptic Problems by Substructuring. , 2010 .

[15]  G. Choudury,et al.  Fully discrete Galerkin approximations of parabolic boundary-value problems with nonsmooth boundary data , 1990 .

[16]  Claes Johnson.  Numerical solution of partial differential equations by the finite element method , 1988 .

[17]  Kenneth Eriksson,et al.  Time discretization of parabolic problems by the discontinuous Galerkin method , 1985 .

[18]  Claes Johnson,et al.  Error estimates and automatic time step control for nonlinear parabolic problems, I , 1987 .

[19]  Irena Lasiecka.  Convergence Estimates for Semidiscrete Approximations of Nonselfadjoint Parabolic Equations , 1984 .