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An adaptive finite element method for steady and transient problems. (English) Zbl 0626.65130

Authors’ summary: Distributing integral error uniformly over variable subdomains, or finite elements, is an attractive criterion by which to subdivide a domain for the Galerkin finite element method when localized steep gradients and high curvatures are to be resolved. Examples are fluid interfaces, shock fronts and other internal layers, as well as fluid mechanical and other boundary layers, e.g. thin-film states at solid walls.
The uniform distribution criterion is developed here into an adaptive technique for one-dimensional problems. Nodal positions can be updated simultaneously with nodal values during Newton iteration, but it is usually better to adopt nearly optimal nodal positions during Newton iteration upon nodal values. Three illustrative problems are solved: steady convection with diffusion, gradient theory of fluid wetting on a solid surface and Buckley-Leverett theory of two-phase Darcy flow in porous media. The new adaptive technique resists entanglement of the nodes of the nodal mesh without requiring the special restrictions upon which the earlier moving finite element method relies.
Reviewer: D.R.Westbrook

MSC:

65Z05 Applications to the sciences
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
76R10 Free convection
76S05 Flows in porous media; filtration; seepage
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