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Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method. (English) Zbl 1195.65170

The aim of the article is to provide a mixed finite element approximation for the Monge-Ampère problem. The authors suggest a vanishing moment method which consist of a quasilinear fourth order problem with a singularly perturbed parameter \(\varepsilon>0\). It is first justified that for each \(\varepsilon>0\), the stated quasilinear fourth order problem has a unique solution \(u^\varepsilon\). It is proved that \(u^\varepsilon\) converges uniformly towards the exact viscosity solution of Monge–Ampère problem, as \(\varepsilon\to 0\). The quasilinear fourth order problem is equivalent to a nonlinear system of second-order equations.
Using this system, the authors derive a weak mixed formulation to the quasilinear fourth order problem in which the solution is denoted by \((u^\varepsilon,\sigma^\varepsilon)\). Thanks to the use of Hermann–Miyoshi mixed elements, the authors suggest a finite element scheme in which the solution is denoted by \((u^\varepsilon_h,\sigma^\varepsilon_h)\). The existence, uniqueness, and the convergence \((u^\varepsilon_h,\sigma^\varepsilon_h)\) towards \((u^\varepsilon,\sigma^\varepsilon)\) is proved, using fixed point technique (since the discrete problem is nonlinear), under the assumption that the mesh parameter \(h\) is small enough and under a regularity assumption on \((u^\varepsilon,\sigma^\varepsilon)\). Finally, the authors present numerical tests showing the error estimates when the mesh parameter \(h\) is a power of \(\varepsilon\).

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J96 Monge-Ampère equations
35J62 Quasilinear elliptic equations
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