Adaptive Tensor Product Grids for Singular Problems

We consider piecewise polynomial approximation of order M in N-dimensions with a tensor product partition of the space. We assume that the partition is to be chosen to minimize ------'the-maximum----em>r---in-the-approximation.------1be---optimal---Iate---gf--cgrwetgence-foyieGeWisef--------polynomial approximation to a smooth function for unconstrained partitions is known to be order ~/N where K is the number of elements in the partition. This rate of convergence is achieved K by a uniform grid which may be taken to be a tensor product. In 1979 de Boor and Rice gave an adaptive algorithm which achieves this same order of convergence for a wide variety of singular functions. We now study whether this optimal order of convergence can be achieved by partitions constrained to be tensor products. We show that the optimal order of convergence is achieved by tensor product grids (partitions) for functions with point or boundary layer singularities. For some other singularities, the tensor product constraint reduces the order of convergence substantially_ This wol1r.was supported in part by the National Science POODdation graDtMS-83-01589