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On the finite element analysis of problems with nonlinear Newton boundary conditions in nonpolygonal domains. (English) Zbl 1066.65124

Poisson’s equation with nonlinear boundary conditions in a bounded two-dimensional domain is considered. The weak solution of this problem is approximated by means of linear triangular finite elements. Using the concept of Zlámal’s ideal element, the authors examine the so-called variational crimes; namely numerical integration and approximation of a piecewise curved boundary by a polygonal one. They prove the existence and uniqueness of the discrete solution, and also the convergence of the proposed finite element method in \(H^1\)-norm without any additional regularity assumptions on the true solution.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J65 Nonlinear boundary value problems for linear elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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References:

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