Systems of Fully Nonlinear Degenerate Elliptic Obstacle problems with Dirichlet boundary conditions

In this paper we prove existence and uniqueness of viscosity solutions of elliptic systems associated to fully nonlinear operators for minimization problems that involve interconnected obstacles. This system appears, among other, in the theory of the so-called optimal switching problems on bounded domains.Comment: Minor corrections throughout the tex

Optimal low-dispersion low-dissipation LBM schemes for computational aeroacoustics

Lattice Boltmzmann Methods (LBM) have been proved to be very effective methods for computational aeroacoustics (CAA), which have been used to capture the dynamics of weak acoustic fluctuations. In this paper, we propose a strategy to reduce the dispersive and disspative errors of the two-dimensional (2D) multi-relaxation-time lattice Boltzmann method (MRT-LBM). By presenting an effective algorithm, we obtain a uniform form of the linearized Navier-Stokes equations corresponding to the MRT-LBM in wave-number space. Using the matrix perturbation theory and the equivalent modified equation approach for finite difference methods, we propose a class of minimization problems to optimize the free-parameters in the MRT-LBM. We obtain this way a dispersion-relation-preserving LBM (DRP-LBM) to circumvent the minimized dispersion error of the MRT-LBM. The dissipation relation precision is also improved.And the stability of the MRT-LBM with the small bulk viscosity is guaranteed. Von Neuman analysis of the linearized MRT-LBM is performed to validate the optimized dispersion/dissipation relations considering monochromatic wave solutions. Meanwhile, dispersion and dissipation errors of the optimized MRT-LBM are quantitatively compared with the original MRT-LBM . Finally, some numerical simulations are carried out to assess the new optimized MRT-LBM schemes.Comment: 33 page

Numerical Solution of the Two-Phase Obstacle Problem by Finite Difference Method

In this paper we consider the numerical approximation of the two-phase membrane (obstacle) problem by finite difference method. First, we introduce the notion of viscosity solution for the problem and construct certain discrete nonlinear approximation system. The existence and uniqueness of the solution of the discrete nonlinear system is proved. Based on that scheme, we propose projected Gauss-Seidel algorithm and prove its convergence. At the end of the paper we present some numerical simulations.Comment: Free Boundary Problem, Two-Phase Membrane Problem, Two-Phase Obstacle Problem, Finite Difference Metho

Heat flows and related minimization problem in image restoration

AbstractA new anisotropic diffusion model is proposed for image restoration and segmentation, which is closely related to the minimization problems for the unconstrained total variation E(u) = ∫Ω α(x)|βu| + (β2)|u − I|2. Existence, uniqueness, and stability of the viscosity solutions of the equation are proved. The experimental results are given and compared with the existing models in the framework of image restoration. The improvement on preserving sharp edges by using the new model is visible