Creation and annihilation of nodes for the moving finite element method
Andrew P. Kuprat · 1992
The Moving Finite Element method and its more geometrically based improvement, Gradient Weighted Moving Finite Elements (GWMFE), approximate the true solution of a PDE using piecewise linear (PL) functions with moving nodes. GWMFE has proved to be extremely efficient at moving its nodes around locally to where they are needed. However, there are problems which can lead to a gradual over-accumulation or under-accumulation of nodes in certain regions, and GWMFE is much less efficient at shifting nodes globally as needed from one region to another. We therefore develop techniques to occasionally create and annihilate nodes as needed by the solution graph. An algorithm is developed which incorporates creation and annihilation 'modules' which assess whether the gridding of a given PL manifold is 'acceptable' or whether the creation or annihilation of nodes is called for. The modules are designed to (1) interact 'fruitfully' with each other, (2) produce an output gridding that is acceptable even if later perturbed by small amounts, (3) keep deformation of the manifold (due to annihilations) to within user-specified tolerances. For the two-dimensional PL manifold case, a 'Local Optimization Procedure' originally introduced by Lawson to achieve Delaunay triangulations of planar point sets is modified to achieve acceptable triangulations of non-planar polygonal regions resulting from node annihilations. We give numerical results for a variety of GWMFE trials in one and two space dimensions, including some quite nontrivial computations for colliding and reflecting shocks of the shallow water equation in 2-D. These techniques promise to make GWMFE in 1-D and 2-D much more efficient and robust. These techniques in computational geometry for 'massaging' a PL manifold should have a great variety of other applications.