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Solvability and Galerkin approximations of a class of nonlinear operator equations. (English) Zbl 1024.65044

The author generates the Babuška-Brezzi theory to a class of nonlinear variational problems with constraints. He considers the operator equations with a dual-dual type structure,

AB * B0(t,σ) u= 𝒢,

where (,𝒢) is given, the operator A itself presents also the dual structure A=A 1 B 1 * B 1 0, B and B 1 are linear and bounded, and A 1 is nonlinear. He provides sufficient conditions for the existence and uniqueness of solutions to the continuous and Galerkin formulations and derives a Strang-type estimate for the associated error. Moreover, he presents an application to the coupling of mixed finite elements and boundary elements for a nonlinear transmission problem in potential theory.

65J15Equations with nonlinear operators (numerical methods)
47J25Iterative procedures (nonlinear operator equations)
35J65Nonlinear boundary value problems for linear elliptic equations
65N15Error bounds (BVP of PDE)
65N50Mesh generation and refinement (BVP of PDE)
65N38Boundary element methods (BVP of PDE)