• Summary
• Related Documents
  • Active Bibliography
  • Co-citation
• Version History

Abstract:

We present a new class of fast solvers for ordinary and partial dierential equations using nite dierence discretizations with adaptive mesh renement. The method is a combination of fast solvers for uniform grids, potential theory, and multilevel sweeps to evaluate layer potentials. In brief, we solve uncoupled problems on uniform data structures and then use layer potentials to patch together solutions at coarse / ne interfaces. Unlike hierarchical iterations, such as multigrid and domain decomposition, the method we propose is direct. For strongly elliptic problems, where the norm of the Green's function kGk is small, one \N"-Cycle gives the optimal result obtainable for a given nite dierence discretization. An \N"-Cycle consists of one downward sweep, from the coarsest to the nest mesh, using boundary conditions inherited from the parent, one upward sweep to compute the jumps at all coarse / ne interfaces, and a second downward sweep to propagate the layer potential correction to all subgrids. Our method preserves the order of accuracy of the underlying dierence scheme without the need for complex stencils at grid boundaries. We illustrate the performance of the method with numerical examples arising in sti and nonsti twopoint boundary value problems and parabolic partial dierential equations in one space dimension.

Citations