65M50 Mesh generation and refinement
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A new adaptive approach for one-dimensional scalar conservation laws with convex flux is proposed. The initial data are approximated on an adaptive grid by a problem dependent, monotone interpolation procedure in such a way, that the multivalued problem of characteristic transport can be easily and explicitly solved. The unique entropy solution is chosen by means of a selection criterion due to LAX. For arbitrary times, the solutions is represented by an adaptive monotone spline interpolation. The spatial approximation is controlled by local $L^1$-error estimated. As a distinctive feature of the approach, there is no discretization in time. The method is monotone on fixed grids. Numerical examples are included, to demonstrate the predicted behavior. {\bf Key words.} method of characteristics, adaptive grids, monotone interpolation, $L^1$-error estimates {\bf AMS(MOS) subject classification.} 65M15, 65M25, 65M50.
Part III of the paper is devoted to the construction of an adaptive FEM solver in two spatial dimensions, which is able to handle the singularly perturbed elliptic problems arising from discretization in time. The problems of error estimation and multilevel iterative solution of the linear systems - both uniformly well behaved with respect to the time step - can be solved simultaneously within the framework of preconditioning. A multilevel nodal basis preconditioner able to handle highly nonuniform meshes is derived. As a numerical example an application of the method to the bioheat-transfer equation is included. {\bf AMS CLASSIFICATION:} 65F10, 65F35, 65M50, 65M60, 65N30.