This work addresses a theory of convergence for finite volume methods applied to linear equations. A non-consistent model problem posed in an abstract Banach space is proved to be convergent. Then various examples show that the functional framework is non-empty. Convergence with a rate h 1 2 of all TVD schemes for linear advection in 1D is an application of the general result. Using duality tec...

A minimax method for finding multiple critical points in Banach spaces is successfully developed in [12] by using a projected pseudo-gradient as a search direction. Since several different techniques can be used to compute a projected pseudo-gradient, the uniform stepsize and the continuity of a search direction, two key properties for the convergence results in [5], get lost. In this paper, in...

We present (geometric) multigrid methods for isogeometric discretization of scalar second order elliptic problems. The smoothing property of the relaxation method, and the approximation property of the intergrid transfer operators are analyzed. These properties, when used in the framework of classical multigrid theory, imply uniform convergence of two-grid and multigrid methods. Supporting nume...

We consider a second order differential operator A( ) = − ∑d i,j=1 ∂iaij( )∂j + ∑d j=1 ∂j ( bj( )· ) +c( ) on R, on a bounded domain D with Dirichlet boundary conditions on ∂D, under mild assumptions on the coefficients of the diffusion tensor aij . The object is to construct monotone numerical schemes to approximate the solution of the problem A( )u( ) = μ( ), ∈ D, where μ is a positive Radon ...

It is known that the complementarity problems and the variational inequality problems are reformulated equivalently as a vector equation by using the natural residual or Fischer-Burmeister function. In this short paper, we first study the global convergence of a sequential injective algorithm for weakly univalent vector equation. Then, we apply the convergence analysis to the regularized smooth...

Convergence to equilibrium of underdamped Langevin dynamics is studied under general assumptions on the potential $U$ allowing for singularities. By modifying direct approach convergence in $L^2$ pioneered by F. H\'erau and developped Dolbeault, Mouhot Schmeiser, we show that converges exponentially fast topologies $L^2(d\mu)$ $L^2(W^* d\mu)$, where $\mu$ denotes invariant probability measure $...

In this article we establish a double-orthogonal principle, and local min-orthogonal method with its step size rule, convergence under assumptions more general than those in previous versions. With such framework, justify mathematically the two new algorithms proposed for solving W-type problems. Numerical examples finding multiple solutions to mixed M-W-type problems illustrate flexibility of ...

The long time numerical approximation of the parabolic p-Laplacian problem with a time-independent forcing term and sufficiently smooth initial data is studied. Convergence and stability results which are uniform for t ∈ [0,∞) are established in the L2, W 1,p norms for the backward Euler and the Crank–Nicholson schemes with the finite element method (FEM). This result extends the existing unifo...

165 In this paper, we wish to analyze the convergence behavior of a number of neuronal plasticity models. Recent neurophysiological research suggests that the neuronal behavior is adaptive. In particular, memory stored within a neuron is associated with the synaptic weights which are varied or adjusted to achieve learning. A number of adaptive neuronal models have been proposed in the literatur...