10 research outputs found
On the rate of convergence of Yosida approximation for rhe nonlocal Cahn-Hilliard equation
It is well-known that one can construct solutions to the nonlocal
Cahn-Hilliard equation with singular potentials via Yosida approximation with
parameter . The usual method is based on compactness arguments
and does not provide any rate of convergence. Here, we fill the gap and we
obtain an explicit convergence rate . The proof is based on the
theory of maximal monotone operators and an observation that the nonlocal
operator is of Hilbert-Schmidt type. Our estimate can provide convergence
result for the Galerkin methods where the parameter could be linked
to the discretization parameters, yielding appropriate error estimates
An implicit midpoint spectral approximation of nonlocal Cahn--Hilliard equations
The paper is concerned with the convergence analysis of a numerical method for nonlocal Cahn-Hilliard equations. The temporal discretization is based on the implicit midpoint rule and a Fourier spectral discretization is used with respect to the spatial variables. The sequence of numerical approximations in shown to be bounded in various norms, uniformly with respect to the discretization parameters, and optimal order bounds on the global error of the scheme are derived. The uniform bounds on the sequence of numerical solutions as well as the error bounds hold unconditionally, in the sense that no restriction on the size of the time step in terms of the spatial discretization parameter needs to be assumed
An implicit midpoint spectral approximation of nonlocal Cahn--Hilliard equations
The paper is concerned with the convergence analysis of a numerical method for nonlocal Cahn--Hilliard equations. The temporal discretization is based on the implicit midpoint rule and a Fourier spectral discretization is used with respect to the spatial variables. The sequence of numerical approximations in shown to be bounded in various norms, uniformly with respect to the discretization parameters, and optimal order bounds on the global error of the scheme are derived. The uniform bounds on the sequence of numerical solutions as well as the error bounds hold unconditionally, in the sense that no restriction on the size of the time step in terms of the spatial discretization parameter needs to be assumed