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A posteriori error estimation techniques for nonlinear elliptic and parabolic PDE’s. (English) Zbl 0948.65092

With the paper in hand, a short but instructive overview of methods leading to a posteriori error estimates for the finite element approximation of nonlinear elliptic and parabolic problems is given. The author, who is one of the leading mathematicians working in this field, also discusses some open problems. In particular, he mentions the quantification of appearing constants, the dependence on problem parameters (as it is of importance for singular perturbed problems), and anisotropies.
Starting with a general quasilinear problem of second-order with homogeneous Dirichlet boundary, the equivalence of the error (measured in the energy norm) and the residual (measured in the dual norm) is shown. For simplicity, the presentation restricts to the usual Hilbert space setting \((H^1_0(\Omega))\) and finite elements being affinely equivalent to simplices or cubes. The family of partitions has, in addition, to fulfill some shape regularity (quasi-uniformity). However, this leads to a problem when anisotropic meshes are needed. Essential for the analysis is the assumption that the Fréchet derivative of the nonlinear differential operator possesses a bounded inverse.
For a simple singular perturbed problem, the right choice of the norm or the dependence on the perturbation parameter, resp., is discussed.
In a next section, a residual based error estimator is constructed and its reliability, efficiency, and locality is shown. Robustness (in the case of a singular perturbation) is also studied.
In a further section, an estimator based upon auxiliary local problems is considered. For details and proofs, the reader should consult the author’s book [A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner, Stuttgart (1986; Zbl 0853.65108)].
Finally, the author turns to the time-dependent case. The functional setting relies, as usual, upon a Gelfand (or evolution) triple of function spaces. For the approximation, a space-time finite element method is employed which allows to use the same framework as for the stationary problem.
The relation to the well-known single step \(\theta\)-scheme as well as to Runge-Kutta time and discontinuous Galerkin discretizations is also shown. However, the method under consideration differs from the discontinuous Galerkin method as the latter one is a non-conforming method leading to difficulties in the error analysis.
Again, error estimators based upon the residual or auxiliary local problems are derived and analyzed.
Due to its clear and well-structured style, the paper seems to be suited for a quick start and can be recommended for those who wish to get acquainted with a posteriori error estimates. At the same time, it gives a short description of the state-of-the-art, completed by a selected list of references.
Reviewer: E.Emmrich (Berlin)

MSC:

65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35K55 Nonlinear parabolic equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Citations:

Zbl 0853.65108
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