Summary
In an abstract framework we present a formalism which specifies the notions of consistency and stability of Petrov-Galerkin methods used to approximate nonlinear problems which are, in many practical situations, strongly nonlinear elliptic problems. This formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems. This theory is illustrated with the example: −div (k(u)▽u) + c • ▽u = f in a two dimensional domain Ω with Dirichlet boundary conditions.
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Pousin, J., Rappaz, J. Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems. Numer. Math. 69, 213–231 (1994). https://doi.org/10.1007/s002110050088
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DOI: https://doi.org/10.1007/s002110050088