A Local Refinement Finite-Element Method for One-Dimensional Parabolic Systems

Abstract

Vector systems of parabolic partial differential equations in one space dimension are solved by an adaptive local mesh refinement Galerkin finite-element procedure. Piecewise linear polynomials are used for the spatial representation of the solution and the backward Euler method is used for temporal integration. A local error indicator based on an estimate of the local discretization error is used to control an adaptive-feedback mesh refinement strategy, where finer space-time meshes are recursively added to coarser ones in regions where greater solution resolution is needed. A posteriori estimates of the local discretization error are obtained by a p-refinement procedure that uses piecewise quadratic hierarchic finite-element approximations in space and trapezoidal rule integration in time. Superconvergence properties of the finite-element method are used to neglect errors at nodes and thereby increase the computational efficiency of the error estimation procedure. Further improvements in computational efficiency are realized by calculating the trapezoidal rule solution as a defect correction to the backward Euler solution. For a model scalar linear parabolic problem, finite-element solutions computed as described above are shown to converge per timestep in $H^1 $ as a uniform space-time mesh is refined. For this problem superconvergence is verified, and global spatial error estimates computed by the above p-refinement procedure are shown to converge to the true error per timestep for uniform refinement. Several computational examples demonstrate the performance of the adaptive mesh refinement procedure for linear and nonlinear problems. Convergence of the error estimate is demonstrated for a linear heat conduction problem.