Volume 23, Issue 1

A multipoint boundary value problem is considered. The existence and uniqueness of solution is proved. Then, for the numerical solution, a general collocation method is proposed.Numerical experiments confirm theoretical results.

In this work, numerical algebraic operations are performed by using several libraries whose algorithm are optimized to drain resources from hardware architecture. In particular, dot product of two vectors and the matrix-matrix product of two dense matrices are computed. In addition, the Cholesky decomposition on a real, symmetric, and positive definite matrix is performed through routines for band and sparse matrix storage. The involved CPU time is used as an indicator of the performance of the employed numerical tool. Results are compared to naive implementations of the same numerical algorithm, highlighting the speed-up due to the usage of optimized routines.

In this work, we are concerned with the finite element approximation for the stationary power law Stokes equations driven by nonlinear slip boundary conditions of ‘friction type’. After the formulation of the problem as mixed variational inequality of second kind, it is shown by application of a variant of Babuska-Brezzi’s theory for mixed problems that convergence of the finite element approximation is achieved with classical assumptions on the regularity of the weak solution. Next, solution algorithm for the mixed variational problem is presented and analyzed in details. Finally, numerical simulations that validate the theoretical findings are exhibited.

In this research using properties of Chebyshev polynomialswe explicitly determine the best uniform polynomial approximation of some classes of functions. In this way we present some new theorems about the best approximation of these classes.

A multigrid method for the Stokes system discretized with an H div -conforming discontinuous Galerkin method is presented. It acts on the combined velocity and pressure spaces and thus does not need a Schur complement approximation. The smoothers used are of overlapping Schwarz type and employ a local Helmholtz decomposition. Additionally, we use the fact that the discretization provides nested divergence free subspaces. We present the convergence analysis and numerical evidence that convergence rates are not only independent of mesh size, but also reasonably small.

It is well-known that the remaining term of a classical n-point Newton-Cotes quadrature depends on at least an n-order derivative of the integrand function. Discounting the fact that computing an n-order derivative requires a lot of differentiation for large n, the main problem is that an error bound for an n-point Newton-Cotes quadrature is only relevant for a function that is n times differentiable, a rather stringent condition. In this paper, by defining two specific linear kernels, we resolve this problem and obtain new error bounds for all closed and open types of Newton-Cotes quadrature rules. The advantage of the obtained bounds is that they do not depend on the norms of the integrand function and are very general such that they cover almost all existing results in the literature. Some illustrative examples are given in this direction.

An optimal bilinear control problem governed by time-harmonic eddy current equations is considered to estimate the electric conductivity of a 3D bounded isotropic domain. The model problem is mainly complicated by the possible presence of non-conducting materials in the domain. We introduce an optimal control approach based on grad-div regularization and divergence penalization. The estimation for the electric conductivity obtained by solving the optimal control problem is allowed to be discontinuous. Here, no higher regularity property can be derived from the corresponding optimality conditions. We analyze the approach and present various numerical results exhibiting its numerical performance