Grid modification for second-order hyperbolic problems

Yang, Dao Qi

doi:10.1090/S0025-5718-1995-1308463-0

Grid modification for second-order hyperbolic problems
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Math. Comp. 64 (1995), 1495-1509 Request permission

Abstract:

A family of Galerkin finite element methods is presented to accurately and efficiently solve the wave equation that includes sharp propagating wave fronts. The new methodology involves different finite element discretizations at different time levels; thus, at any time level, relatively coarse grids can be applied in regions where the solution changes smoothly while finer grids can be employed near wave fronts. The change of grid from time step to time step need not be continuous, and the number of grid points at different time levels can be arbitrarily different. The formulation is applicable to general second-order hyperbolic equations. Stability results are proved and a priori error estimates are established for several boundary conditions. Our error estimates consist of three parts: the time finite difference discretization error, the spatial finite element discretization error, and the error due to the projections of the approximated solution from old grids onto new grids.

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