Multiscale scattering in nonlinear Kerr-type media
Copyright policy )Multiscale scattering in nonlinear Kerr-type media
We propose a multiscale approach for a nonlinear Helmholtz problem with possible oscillations in the Kerr coefficient, the refractive index, and the diffusion coefficient. The method does not rely on structural assumptions on the coefficients and combines the multiscale technique known as Localized Orthogonal Decomposition with an adaptive iterative approximation of the nonlinearity. We rigorously analyze the method in terms of well-posedness and convergence properties based on suitable assumptions on the initial data and the discretization parameters. Numerical examples illustrate the theoretical error estimates and underline the practicability of the approach.
- Friedrich Schiller University Jena Germany
- Chalmers University of Technology Sweden
- University of Gothenburg Sweden
Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation, a priori estimates, Kerr medium, Existence problems for PDEs: global existence, local existence, non-existence, Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness, Lasers, masers, optical bistability, nonlinear optics, Numerical Analysis (math.NA), Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, Stability and convergence of numerical methods for boundary value problems involving PDEs, A priori estimates in context of PDEs, 65N12, 65N30, 35G30, multiscale method, FOS: Mathematics, nonlinear, Helmholtz equation, Mathematics - Numerical Analysis, PDEs in connection with optics and electromagnetic theory, Boundary value problems for nonlinear higher-order PDEs
Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation, a priori estimates, Kerr medium, Existence problems for PDEs: global existence, local existence, non-existence, Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness, Lasers, masers, optical bistability, nonlinear optics, Numerical Analysis (math.NA), Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, Stability and convergence of numerical methods for boundary value problems involving PDEs, A priori estimates in context of PDEs, 65N12, 65N30, 35G30, multiscale method, FOS: Mathematics, nonlinear, Helmholtz equation, Mathematics - Numerical Analysis, PDEs in connection with optics and electromagnetic theory, Boundary value problems for nonlinear higher-order PDEs
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