Adaptive refinement and nonlinear fluid problems

Abstract

Abstract Two principal and related topics are considered: (1) adaptive mesh refinement for finite element computations, and (2) mesh refinement specifically for nonlinear flow problems. In the first instance the residual and associated trace theorems for variational problems are introduced to relate the solution error to a computable residual. This provides a theoretical basis for a mesh refinement strategy. A corresponding adaptive refinement procedure that automatically and selectively refines the mesh is formulated and implemented for a class of nonlinear transport problems in chemical engineering. In particular, a nonlinear problem with a boundary-layer solution is investigated. The strategy of interweaving Newton solution and mesh refinement proves particularly efficient. Two-dimensional compressible and transonic flows are next examined. Mesh refinement of subrogions of the flow field is applied to yield high solution accuracy near a singularity for both the linear and nonlinear flows. Refinement and Newton iteration are combined, together with Mach number parameterization, to determine an efficient and accurate solution algorithm. Similar points for nonlinear viscous problems are also reviewed.