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Numerical analysis of strongly nonlinear PDEs. (English) Zbl 1381.65092

This is an impressive review about the stand of the art in the numerical approximation of boundary value problems for nonlinear second-order convex and non-convex partial differential equations (PDEs). The intention is to summarize the 25 years of development of fully nonlinear numerical PDEs. On 167 pages, each one full of mathematics, an immense variety of results is presented. The following table of contents which gives a good idea of the topics covered:
\(\bullet\)
Section 1. Introduction (12 p.)
\(\bullet\)
Section 2. Elements of the theory of strongly nonlinear PDEs (32 p.)
\(\bullet\)
Section 3. Monotonicity in numerical methods (31 p.)
\(\bullet\)
Section 4. Finite element methods for elliptic problems in non-divergence form (20 p.)
\(\bullet\)
Section 5. Discretization of convex second-order elliptic equations (22 p.)
\(\bullet\)
Section 6. The Monge-Ampère equation (25 p.)
\(\bullet\)
Section 7. Discretizations of non-convex second-order elliptic equations (22 p.)
\(\bullet\)
References. 135 items

A few additional remarks may be in order. Throughout the paper for many of the presented results also proofs (or at least an indication of the main steps in a prospective proof) are given. This is, naturally, not the case in Section 2, where the basic theory and analysis of elliptic PDEs is reviewed. This section is quite useful since these results motivate to a good extent the construction and analysis of the numerical methods. Here also the important notion of viscosity solutions is introduced which uses to serve as the adequate concept when studying the convergence of numerical schemes. In Section 3 the convergence results by G. Barles and P. E. Souganidis [Asymptotic Anal. 4, No. 3, 271–283 (1991; Zbl 0729.65077)] for consistent, stable and monotone approximation schemes satisfying a comparision principle play a central role. A lot of material about positivity, maximum principles (among them Alexandrov-Bakelman-Pucci), comparision for finite differences and finite elements is presented. More on finite elements, now applied to linear problems in non-divergence form with non-smooth coefficients, can be found in Section 4. The Cordes condition is introduced. A discrete Miranda-Talenti estimate and a discrete Calderón-Zygmund estimate is proved. In the next two sections the so far discussed results to the Hamilton-Jacobi-Bellmann and the Monge-Ampère equation with emphasis on obtaining rates of convergence. For the latter equation the appplication of semi-Lagrangian schemes are illustrated. The solution of the discrete problems is also given some attention. Finally, Section 7 is concerned with discretizations for non-convex elliptic equations. The paper closes with a short outlook on open problems.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
35J57 Boundary value problems for second-order elliptic systems
35J66 Nonlinear boundary value problems for nonlinear elliptic equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35J96 Monge-Ampère equations
35F21 Hamilton-Jacobi equations

Citations:

Zbl 0729.65077
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Full Text: DOI arXiv

References:

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