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Stabilized and inexact adaptive methods for capturing internal layers in quasilinear PDE. (English) Zbl 1346.65051

Summary: A method is developed within an adaptive framework to solve quasilinear diffusion problems with internal and possibly boundary layers starting from a coarse mesh. The solution process is assumed to start on a mesh where the problem is badly resolved, and approximation properties of the exact problem and its corresponding finite element solution do not hold. A sequence of stabilized and inexact partial solves allows the mesh to be refined to capture internal layers while an approximate solution is built eventually leading to an accurate approximation of both the problem and its solution. The innovations in the current work include a closed form definition for the numerical dissipation and inexact scaling parameters on each mesh refinement, as well as a convergence result for the residual of the discrete problem. Numerical experiments demonstrate the method on a range of problems featuring steep internal layers and high solution dependent frequencies of the diffusion coefficients.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K59 Quasilinear parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs

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References:

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