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Peter K. Moore : An Adaptive H-Refinement Finite Element For Solving Systems of Parabolic Partial Differential Equations in Three Space Dimensions

Adaptive methods for solving systems of partial differential equations have become widespread. Robust adaptive software for solving parabolic systems in one and two space dimensions is now widely available. Three spatial adaptive strategies and combinations thereof are frequently employed: mesh refinement (h-refinement); mesh motion (r-refinement); and order variation (p-refinement). These adaptive strategies are driven by a priori and a posteriori error estimates. I will present an adaptive h-refinement finite element code in three dimensions on structured grids. These structured grids contain irregular nodes. Solution values at these nodes are determined by continuity requirements across element boundaries rather than by the differential equations. The differential-algebraic system resulting from the spatial discretization is integrated using Linda Petzold's multistep DAE code DASPK. The large linear systems resulting from Newton's method applied to nonlinear system of differential algebraic equations is solved using preconditioned GMRES. In DASPK the matrix-vector products needed by GMRES are approximated by a ``directional derivative''. Thus, the Jacobian matrix need not be assembled. However, this approach is inefficient. I have modified DASPK to compute the matrix-vector product using stored Jacobian matrix. As in the earlier version of DASPK, DASSL, this matrix is kept for several time steps before being updated. I will discuss appropriate preconditioning strategies, including fast-banded preconditioners. In three dimensions when using multistep methods for time integration it is crucial to use a ``warm restart'', that is, to restart the dae solver at the current time step and order. This requires interpolation of the history information. The interpolation must be done in such a way that mode irregularity is enforced on the new grid. A posteriori error estimates on uniform grids can easily be generalized from two-dimensional results (Babuska and Yu showed that in the case of odd order elements, jumps across elemental boundaries give accurate estimates, and in the case of even order elements, local parabolic systems must be solved to obtain accurate estimates). Babuska's work can even be generalized to meshes with irregular modes but now they no longer converge to the true error (in the case of odd order elements). I have developed a new set of estimates that extend the work of Babuska to irregular meshes and finite difference methods. These estimates provide a posteriori error indicators in the finite element context. Several examples that demonstrate the effectiveness of the code will be given.

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