Error estimates for abstract linear–quadratic optimal control problems using proper orthogonal decomposition

In this paper we investigate POD discretizations of abstract linear–quadratic optimal control problems with control constraints. We apply the discrete technique developed by Hinze (Comput. Optim. Appl. 30:45–61, 2005) and prove error estimates for the corresponding discrete controls, where we combine error estimates for the state and the adjoint system from Kunisch and Volkwein (Numer. Math. 90:117–148, 2001; SIAM J. Numer. Anal. 40:492–515, 2002). Finally, we present numerical examples that illustrate the theoretical results.     

Numerical experiments on the condition number of the interpolation matrices for radial basis functions

Through numerical experiments, we examine the condition numbers of the interpolation matrix for many species of radial basis functions (RBFs), mostly on uniform grids. For most RBF species that give infinite order accuracy when interpolating smooth f(x)—Gaussians, sech's and Inverse Quadratics—the condition number κ(α,N) rapidly asymptotes to a limit κasymp(α) that is independent of N and depends only on α, the inverse width relative to the grid spacing. Multiquadrics are an exception in that the condition number for fixed α grows as N². For all four, there is growth proportional to an exponential of 1/α (1/α² for Gaussians). For splines and thin-plate splines, which contain no width parameter, the condition numbers grows asymptotically as a power of N—a large power as the order of the RBF increases. Random grids typically increase the condition number (for fixed RBF width) by orders of magnitude. The quasi-random, low discrepancy Halton grid may, however, have a lower condition number than a uniform grid of the same size.     

Preconditioning for radial basis functions with domain decomposition methods

In our previous work, an effective preconditioning scheme that is based upon constructing least-squares approximation cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. The preconditioner costs O(N²) flops to set up and O(N) storage. The preconditioning technique is sufficiently general that it can be applied to different types of different operators. This was applied to the 2D multiquadric method, with c~1/√N on the Poisson test problem, the preconditioned GMRES converges in tens of iterations. In this paper, we combine the RBF methods and the ACBF preconditioning technique with the domain decomposition method (DDM). We studied different implementations of the ACBF-DDM scheme and provide numerical results for N > 10,000 nodes. We shall demonstrate that the efficiency of the ACBF-DDM scheme improves dramatically as successively finer partitions of the domain are considered.     

Coupling three-field formulation and meshless mixed Galerkin methods using radial basis functions

In this work, we solve the elliptic partial differential equation by coupling the meshless mixed Galerkin approximation using radial basis function with the three-field domain decomposition method. The formulation has been adopted to increase the efficiency of the numerical technique by decreasing the error and dealing with the ill conditioning of the linear system caused by the radial basis function. Convergence analysis of the coupled technique is treated and numerical results of some solved examples are given at the end of this paper.     

Mixed methods for fourth-order elliptic and parabolic problems using radial basis functions

By extending Wendlands meshless Galerkin methods using RBFs, we develop mixed methods for solving fourth-order elliptic and parabolic problems by using RBFs. Similar error estimates as classical mixed finite element methods are proved. AMS subject classification 35G15, 65N12     

Meshless solution of two-dimensional incompressible flow problems using the radial basis integral equation method

The two-dimensional incompressible fluid flow problems governed by the velocity–vorticity formulation of the Navier–Stokes equations were solved using the radial basis integral (RBIE) equation method. The RBIE is a meshless method based on the multi-domain boundary element method with overlapping subdomains. It solves at each node for the potential and its spatial derivatives. This feature of the RBIE is advantageous in solving the velocity–vorticity formulation of the Navier–Stokes equations since the calculated velocity gradients can be used to compute the vorticity that is prescribed as a boundary condition to the vorticity transport equation. The accuracy of the numerical solution was examined by solving the test problem with known analytical solution. Two benchmark problems, i.e. the lid driven cavity flow and the thermally driven cavity flow were also solved. The numerical results obtained using the RBIE showed very good agreement with the benchmark solutions.     

Higher‐order discontinuous Galerkin method for pyramidal elements using orthogonal bases

We study finite elements of arbitrarily high‐order defined on pyramids for discontinuous Galerkin methods. We propose a new family of high‐order pyramidal finite elements using orthogonal basis functions which can be used in hybrid meshes including hexahedra, tetrahedra, wedges, and pyramids. We perform a comparison between these orthogonal functions and nodal functions for affine and non‐affine elements. Different strategies for the inversion of the mass matrix are also considered and discussed. Numerical experiments are conducted for the three dimensional Maxwell's equations. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Combined effect of magnetic field and heat absorption on unsteady free convection and heat transfer flow in a micropolar fluid past a semi-infinite moving plate with viscous dissipation using element free Galerkin method

The fully developed electrically conducting micropolar fluid flow and heat transfer along a semi-infinite vertical porous moving plate is studied including the effect of viscous heating and in the presence of a magnetic field applied transversely to the direction of the flow. The Darcy-Brinkman-Forchheimer model which includes the effects of boundary and inertia forces is employed. The differential equations governing the problem have been transformed by a similarity transformation into a system of non-dimensional differential equations which are solved numerically by element free Galerkin method. Profiles for velocity, microrotation and temperature are presented for a wide range of plate velocity, viscosity ratio, Darcy number, Forchhimer number, magnetic field parameter, heat absorption parameter and the micropolar parameter. The skin friction and Nusselt numbers at the plates are also shown graphically. The present problem has significant applications in chemical engineering, materials processing, solar porous wafer absorber systems and metallurgy.