Volume 12, Issue 4

It is well known that the solution of the Stokes or Navier–Stokes system in a non convex polygonal domain of has a singular behaviour near non convex corners. Consequently we investigate different refined (non conforming) finite volume-element methods to approximate the solution of such problems and restore optimal orders of convergence as for smooth solutions. Numerical tests are presented, which confirm the theoretical rates of convergence and illustrate the advantage of the use of refined meshes.

We introduce and analyze a multigrid algorithm for higher order finite difference schemes for elliptic problems on a nonuniform rectangular mesh. These schemes are presented by 9-point stencils. We prove the V-cycle convergence adopting the theory developed for finite element methods to these schemes. To be more precise, we show that the energy norm of the prolongation operator is less than one and hence obtain the conclusion using the approximation and regularity property as in [2]. In the numerical experiment section, we report contraction numbers, eigenvalues and condition numbers of the multigrid algorithm. The numerical test shows that for higher order schemes the multigrid algorithm converges much faster than for low order schemes. We also test the case of a nonuniform grid with a line smoother which also shows good convergence behavior.

A new multi-step curve search method for unconstrained minimization problems is proposed. The convergence of the algorithm is proved under some mild conditions. The linear convergence rate is also investigated when the objective function is uniformly convex. This method uses previous multi-step iterative information and curve search rule to generate new iterative points. Using more previous iterative information and curve search rule can make the new method converge more stably than traditional descent methods and be suitable to solve large scale problems.

We study a two-level overlapping additive Schwarz preconditioner for the h -version of the Galerkin boundary element method when used to solve hypersingular integral equations of the first kind on an open surface in These integral equations result from Neumann problems for the Laplace and Lamé equations in the exterior of the surface. We prove that the condition number of the preconditioned system is bounded by O (1 + log 2 ( H / δ )), where H denotes the diameter of the subdomains and δ the size of the overlap.