Convergence of a finite volume scheme for an elliptic-hyperbolic system
Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation
We consider a degenerate parabolic system which models the evolution of nematic liquid crystal with variable degree of orientation. The system is a slight modification to that proposed in [Calderer et al., SIAM J. Math. Anal.33 (2002) 1033–1047], which is a special case of Ericksen's general continuum model in [Ericksen, Arch. Ration. Mech. Anal.113 (1991) 97–120]. We prove the global existence of weak solutions by passing to the limit in a regularized system. Moreover, we propose a practical...
Convergence of a Lagrange-Galerkin method for a fluid-rigid body system in ALE formulation
We propose a numerical scheme to compute the motion of a two-dimensional rigid body in a viscous fluid. Our method combines the method of characteristics with a finite element approximation to solve an ALE formulation of the problem. We derive error estimates implying the convergence of the scheme.
Convergence of a lattice numerical method for a boundary-value problem with free boundary and nonlinear Neumann boundary conditions.
Convergence of a method for solving the magnetostatic field in nonlinear media
For solving the boundary-value problem for potential of a stationary magnetic field in two dimensions in ferromagnetics it is possible to use a linearization based on the succesive approximations. In this paper the convergence of this method is proved under some conditions.
Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics
We prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. The equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, which arises in the theory of dislocation dynamics. We show that if an anisotropic mean curvature motion is approximated by equations of this type then it is always of variational type, whereas the converse is true only in dimension two.
Convergence of a numerical scheme for a nonlinear oblique derivative boundary value problem
We present here a discretization of a nonlinear oblique derivative boundary value problem for the heat equation in dimension two. This finite difference scheme takes advantages of the structure of the boundary condition, which can be reinterpreted as a Burgers equation in the space variables. This enables to obtain an energy estimate and to prove the convergence of the scheme. We also provide some numerical simulations of this problem and a numerical study of the stability of the scheme, which appears...
Convergence of a numerical scheme for a nonlinear oblique derivative boundary value problem
We present here a discretization of a nonlinear oblique derivative boundary value problem for the heat equation in dimension two. This finite difference scheme takes advantages of the structure of the boundary condition, which can be reinterpreted as a Burgers equation in the space variables. This enables to obtain an energy estimate and to prove the convergence of the scheme. We also provide some numerical simulations of this problem and a numerical study of the stability of the scheme, which appears...
Convergence of a proposed adaptive WENO scheme for Hamilton-Jacobi equations
We study high-order numerical methods for solving Hamilton-Jacobi equations. Firstly, by introducing new clear concise nonlinear weights and improving their convex combination, we develop WENO schemes of Zhu and Qiu (2017). Secondly, we give an algorithm of constructing a convergent adaptive WENO scheme by applying the simple adaptive step on the proposed WENO scheme, which is based on the introduction of a new singularity indicator. Through detailed numerical experiments on extensive problems including...
Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions
Convergence of a two-grid algorithm for the control of the wave equation
We analyze the problem of boundary observability of the finite-difference space semidiscretizations of the 2-d wave equation in the square. We prove the uniform (with respect to the meshsize) boundary observability for the solutions obtained by the two-grid preconditioner introduced by Glowinski [9]. Our method uses previously known uniform observability inequalities for low frequency solutions and a dyadic spectral time decomposition. As a consequence we prove the convergence of the two-grid algorithm...
Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation
We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation on an interval. The discretization is based on the equation’s gradient flow structure with respect to the Wasserstein distance. The scheme inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, metric contraction and minimum/ maximum principles. As the main result, we give a proof of convergence in the limit of vanishing mesh size under a CFL-type condition. We...
Convergence of an accurate scheme for first order quasi linear equations
Convergence of an entropic semi-discretization for nonlinear Fokker-Planck equations in Rd
Convergence of an equilibrium finite element model for plane elastostatics
An equilibrium triangular block-element, proposed by Watwood and Hartz, is subjected to an analysis and its approximability property is proved. If the solution is regular enough, a quasi-optimal error estimate follows for the dual approximation to the mixed boundary value problem of elasticity (based on Castigliano's principle). The convergence is proved even in a general case, when the solution is not regular.
Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh.
Convergence of approximate solutions of a differential equation
Convergence of approximate solutions to scalar conservation laws by degenerate diffusion
Convergence of approximation methods for eigenvalue problem for two forms
The paper concerns an approximation of an eigenvalue problem for two forms on a Hilbert space . We investigate some approximation methods generated by sequences of forms and defined on a dense subspace of . The proof of convergence of the methods is based on the theory of the external approximation of eigenvalue problems. The general results are applied to Aronszajn’s method.
Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations.