Band 19, Heft 3

We propose and analyze overlapping two-level additive Schwarz preconditioners for the local discontinuous Galerkin discretization. We prove that the condition number of the preconditioned system is bounded by C [1 + ( H / δ )], where H represents the coarse mesh size, δ measures the overlap among the subdomains, and the constant C is independent of H , δ , the fine mesh size h and the number of subdomains N s . Numerical results are presented showing the scalability of the method.

The formulation as well as full-multigrid solution of control-constrained Cauchy–Riemann optimal control problems are presented. A constrained distributed control mechanism through divergence and curl sources is considered with mixed boundary conditions. The corresponding optimal solutions are obtained solving a Cauchy–Riemann optimality system consisting of four first-order partial differential equations and two inequality constraints. For the solution of the optimality system, staggered grids and a full-multigrid scheme are considered. The proposed full-multigrid method is based on a coarsening strategy by a factor of three that results in a nested hierarchy of staggered grids. The smoothing procedure consists of a distributed-Gauss–Seidel scheme for the state and adjoint equations and a projected gradient step for the controls. Numerical results validate the effectiveness of the proposed approach.

Three one-parameter, eighth-order families of symplectic integrators are presented. They are based on the Strang splitting, the Forest–Ruth construction, and the Zassenhaus formula, respectively. In each time-step, a free parameter is adopted to preserve energy exactly in this step. Numerical experiments show that such a procedure is possible both in double- and quad-precision computations. Therefore, our integrators can be called energy preserving symplectic integrators.

In this paper, a numerical method is developed to compute an approximate solution of high-order linear complex differential equations in elliptic domains. By using collocation points and Bessel polynomials, this method transforms the linear complex differential equation into a matrix equation which corresponds to a system of linear equations in the unknown Bessel coefficients. The suggested method reproduces the analytic solution when it is polynomial. Some numerical examples are given to illustrate the applicability and validity of the technique and comparisons are made with other existing approaches.