An Adaptive Finite Element Method for Parabolic Differential Systems: Some Algorithmic Considerations in Solving in Three Space Dimensions

An adaptive h-refinement method is described for solving systems of parabolic partial differential equations in three space dimensions on hexahedral grids. These grids typically have irregular (hanging) nodes. Solutions are calculated using Galerkin's method with a piecewise trilinear basis in space and a backward difference formula (BDF) code in time. New a posteriori error indicators based on interpolation error estimates for irregular grids are used to control refinement and coarsening. A more efficient algorithm for assembling banded portions of the Jacobian is introduced. A simple strategy for dealing with storage limitations by limiting the level of refinement is developed. Computational results demonstrate the effectiveness of the adaptive method on linear and nonlinear problems.