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Some aspects of adaptive grid technology related to boundary and interior layers. (English) Zbl 1049.65100

Summary: We consider the use of adaptive mesh strategies for solution of problems exhibiting boundary and interior layer solutions. As the presence of these layer structures suggests, reliable and accurate solution of this class of problems using finite difference, finite volume or finite element schemes requires grading the mesh into the layers and due attention to the associated algorithms. When the nature and structure of the layer is known, mesh grading can be achieved during the grid generation by specifying an appropriate grading function.
However, in many applications the location and nature of the layer behavior is not known in advance. Consequently, adaptive mesh techniques that employ feedback from intermediate grid solutions are an appealing approach.
In this paper, we provide a brief overview of the main adaptive grid strategies in the context of problems with layers. Associated error indicators that guide the refinement feedback control/grid optimization process are also covered and there is a brief commentary on the supporting data structure requirements. Some current issues concerning the use of stabilization in conjunction with adaptive mesh refinement (AMR), the question of ”pollution effects” in computation of local error indicators, the influence of nonlinearities and the design of meshes for targeted optimization of specific quantities are considered.
The application of AMR for layer problems is illustrated by means of case studies from semiconductor device transport (drift diffusion), nonlinear reaction–diffusion, layers due to surface capillary effects, and shockwaves in compressible gas dynamics.

MSC:

65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
35K15 Initial value problems for second-order parabolic equations
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
82D37 Statistical mechanics of semiconductors
76N15 Gas dynamics (general theory)
76M10 Finite element methods applied to problems in fluid mechanics

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