Volume 23, Issue 3

In [L. Badea, Global convergence rate of a standard multigrid method for variational inequalities, IMA J. Numer. Anal., 34 (2014), No. 1, 197-216], a global convergence rate of the standard monotone multigrid method for variational inequalities is derived. This algorithm can be also viewed as performing multiplicative iterations on each level and also multiplicative iterations over the levels. In the present paper, this algorithm together with other three algorithms,which are combinations of additive or multiplicative iterations on levels with additive or multiplicative iterations over the levels, are analyzed in a unitary manner and in a more general framework which allow us to consider problems in the Sobolev space W 1, σ , 1 < σ < ∞, not only in the usual H 1 . The algorithms are given for the constrained minimization of functionals where the convex set is of two-obstacle type and have an optimal computing complexity of the iteration steps. The estimations of the global convergence rate are written in function of the number of levels.

The subject of this paper is the analysis of the space-time discontinuous Galerkin method for the solution of nonstationary, nonlinear, convection-diffusion problems. In the formulation of the numerical scheme, the nonsymmetric, symmetric and incomplete versions of the discretization of diffusion terms and interior and boundary penalty are used. Then error estimates are briefly characterized. The main attention is paid to the investigation of unconditional stability of the method. An important tool is the concept of the discrete characteristic function. Theoretical results are accompanied by numerical experiments.

Extended second derivative multistep methods and their modified form have been introduced as efficient numerical solvers for stiff systems. Considering these methods from second derivative general linear methods point of view, we construct some perturbations of these methods which improve their stability properties while preserve their order. Numerical experiments confirm efficiency of the constructed perturbed methods.

The extended linear complementarity problem (denoted by XLCP), of which the linear and horizontal linear complementarity problems are two special cases, can be reformulated as the solution of a nonsmooth system of equations. By the symmetrically perturbed smoothing Fischer-Burmeister function, the XLCP is approximated by a family of parameterized smoothness optimization problems. Asmoothing damped Gauss-Newton method is designed for solving the XLCP. The proposed algorithm is proved to be convergent globally under suitable assumptions. Some numerical results are reported in the paper

In this paper, Hermite and Lagrange interpolation by G 1 smooth cubic spatial splines with small strain energy are considered. A parametric interpolation scheme, based on minimisation of an approximate strain energy, is introduced. As its particular cases, it reproduces three known planar schemes and one spatial. The resulting interpolants are shape preserving, locally without loops, cusps or folds. A construction of optimal tangent directions is analysed. This yields a solution of the Lagrange interpolation problem. The obtained curves are compared with the results of two classic schemes and are applied for the construction of a cubic Hermite spline surface interpolant with small Willmore energy.

The Chang-Cooper discretization scheme for a class of Fokker-Planck equations is investigated. These equations of parabolic type govern the time evolution of the probability density function of stochastic processes, such that positivity of the density function and conservativeness of the total probability is guaranteed. It is shown that the Chang-Cooper scheme combined with backward first- and second-order finite differencing in time provides stable and accurate solutions that are conservative and positive. These properties are theoretically proven and validated by numerical experiments.