Application of a 14-point averaging operator in the grid method

The Dirichlet problem for Laplace’s equation in a rectangular parallelepiped is solved by applying the grid method. A 14-point averaging operator is used to specify the grid equations on the entire grid introduced in the parallelepiped. Given boundary values that are continuous on the parallelepiped edges and have first derivatives satisfying the Lipschitz condition on each parallelepiped face, the resulting discrete solution of the Dirichlet problem converges uniformly and quadratically with respect to the mesh size. Assuming that the boundary values on the faces have fourth derivatives satisfying the Hölder condition and the second derivatives on the edges obey an additional compatibility condition implied by Laplace’s equation, the discrete solution has uniform and quartic convergence with respect to the mesh size. The convergence of the method is also analyzed in certain cases when the boundary values are of intermediate smoothness.     

Application of the spherical metric tensor to grid adaptation and the solution of applied problems

New results concerning the construction and application of adaptive numerical grids for solving applied problems are presented. The grid generation technique is based on the numerical solution of inverted Beltrami and diffusion equations for a monitor metric. The capabilities of the spherical metric tensor as applied to adaptive grid generation are examined in detail. Adaptive hexahedral grids are used to numerically solve a boundary value problem for the three-dimensional heat equation with a moving boundary in a continuous medium with discontinuous thermophysical parameters; this problem models the interaction of a thermal wave with a thermocouple embedded in the solid.     

Multiple grid and Osher's scheme for the efficient solution of the steady Euler equations

An iterative method is developed for the solution of the steady Euler equations for inviscid flow. The system of hyperbolic conservation laws is discretized by a finite-volume Osher-discretization. The iterative method is a multiple grid (FAS) iteration with symmetric Gauss-Seidel (SGS) as a relaxation method. Initial estimates are obtained by full multigrid (FMG). In the pointwise relaxation the equations are kept in block-coupled form and local linearization of the equations and the boundary conditions is considered. The efficient formulation of Osher's discretization of the 2-D non-isentropic steady Euler equations and its linearization is presented. The efficiency of FAS-SGS iteration is shown for a transonic model problem. It appears that, for the problem considered, the rate of convergence is almost independent of the gridsize and that for all meshsizes the discrete system is solved up to truncation error accuracy in only a few (2 or 3) iteration cycles.     

Two‐grid variational multiscale algorithms for the stationary incompressible Navier‐Stokes equations with friction boundary conditions

Two‐grid variational multiscale (VMS) algorithms for the incompressible Navier‐Stokes equations with friction boundary conditions are presented in this article. First, one‐grid VMS algorithm is used to solve this problem and some error estimates are derived. Then, two‐grid VMS algorithms are proposed and analyzed. The algorithms consist of nonlinear problem on coarse grid and linearized problem (Stokes problem or Oseen problem) on fine grid. Moreover, the stability and convergence of the present algorithms are established. Finally, Numerical results are shown to confirm the theoretical analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 546–569, 2017     

The role of conditioning in mesh selection algorithms for first order systems of linear two point boundary value problems

Codes for the numerical solution of two-point boundary value problems can now handle quite general problems in a fairly routine and reliable manner. When faced with particularly challenging equations, such as singular perturbation problems, the most efficient codes use a highly non-uniform grid in order to resolve the non-smooth parts of the solution trajectory. This grid is usually constructed using either a pointwise local error estimate defined at the grid points or else by using a local residual control. Similar error estimates are used to decide whether or not to accept a solution. Such an approach is very effective in general providing that the problem to be solved is well conditioned. However, if the problem is ill conditioned then such grid refinement algorithms may be inefficient because many iterations may be required to reach a suitable mesh on which to compute the solution. Even worse, for ill conditioned problems an inaccurate solution may be accepted even though the local error estimates may be perfectly satisfactory in that they are less than a prescribed tolerance. The primary reason for this is, of course, that for ill conditioned problems a small local error at each grid point may not produce a correspondingly small global error in the solution. In view of this it could be argued that, when solving a two-point boundary value problem in cases where we have no idea of its conditioning, we should provide an estimate of the condition number of the problem as well as the numerical solution. In this paper we consider some algorithms for estimating the condition number of boundary value problems and show how this estimate can be used in the grid refinement algorithm.     

Consistent grid operators with the cell-nodal definition of grid functions

A principle of consistency for grid operators that ensures grid-operator inhomogeneous boundary-value problems are posed well is considered. Grid analogs of first-order differential operators and boundary operators that are consistent in the sense of fulfilling the grid analogs of integral relations are constructed on an irregular triangular grid. These relations are corollaries to the divergence theorem for vector fields that are the product of a scalar by a vector, the vector product of vectors, or the interior product of a vector by a dyadic. In each grid relation, one function is defined at the nodes; the other, in cells. Construction is performed using a grid-operator interpretation of the corollaries to the integral relations, which hold if one of the functions is a piecewise-linear interpolant to the nodal grid function, and the other is the piecewise-constant interpolant of the cell grid function.     

Sparse grids for boundary integral equations

Summary. The potential of sparse grid discretizations for solving boundary integral equations is studied for the screen problem on a square in . Theoretical and numerical results on approximation rates, preconditioning, adaptivity and compression for piecewise constant and linear sparse grid spaces are obtained. Received March 17, 1998 / Revised version received September 10, 1998     

Uniform grid approximation of nonsmooth solutions with improved convergence for a singularly perturbed convection-diffusion equation with characteristic layers

A mixed boundary value problem for a singularly perturbed elliptic convection-diffusion equation with constant coefficients in a square domain is considered. Dirichlet conditions are specified on two sides orthogonal to the flow, and Neumann conditions are set on the other two sides. The right-hand side and the boundary functions are assumed to be sufficiently smooth, which ensures the required smoothness of the desired solution in the domain, except for neighborhoods of the corner points. Only zero-order compatibility conditions are assumed to hold at the corner points. The problem is solved numerically by applying an inhomogeneous monotone difference scheme on a rectangular piecewise uniform Shishkin mesh. The inhomogeneity of the scheme lies in that the approximating difference equations are not identical at different grid nodes but depend on the perturbation parameter. Under the assumptions made, the numerical solution is proved to converge ?-uniformly to the exact solution in a discrete uniform metric at an O(N ^?3/2ln² N) rate, where N is the number of grid nodes in each coordinate direction.     

Solvability of a boundary value problem for a stationary model of the magnetic hydrodynamics of a viscous heat-conducting fluid

A boundary value problem is studied for a stationary model of the magnetic hydrodynamics of a viscous heat-conducting fluid under nonhomogeneous boundary conditions on the velocity, electromagnetic field, and temperature. The model consists of the Navier-Stokes equations, the Maxwell equations, the generalized Ohm law, and the convection-diffusion equation for the temperature which are connected nonlinearly with each other. Sufficient conditions on the initial data are established that guarantee the global solvability of the problem under consideration and the local uniqueness of its solution. The properties are studied of the linear operator obtained by linearizing the operator of the original boundary value problem.     

Study of the contraction flow using grid generation technique

We study a model procedure to solve the incompressible Navier-Stokes equations on the flow inside contraction geometry. The governing equations are expressed in the primitive variable formulation. A rectangular computational plane is arises by elliptic grid generation technique. The numerical solution is based on a technique of automatic numerical generation of a curvilinear coordinate system. By transformed the governing equation into computational plane. The time dependent momentum equations are solved explicitly for the velocity field using the explicit marching procedure, the continuity equation is applied at each grid point in the solution of pressure equation, while the successive over relaxation (SOR) method is used for the Neumann problem for pressure. We will apply the technique on several irregular-shape.     

On a combined grid method for solving the Dirichlet problem for the Laplace equation in a rectangular parallelepiped

A combined grid method for solving the Dirichlet problem for the Laplace equation in a rectangular parallelepiped is proposed. At the grid points that are at the distance equal to the grid size from the boundary, the 6-point averaging operator is used. At the other grid points, the 26-point averaging operator is used. It is assumed that the boundary values have the third derivatives satisfying the Lipschitz condition on the faces; on the edges, they are continuous and their second derivatives satisfy the compatibility condition implied by the Laplace equation. The uniform convergence of the grid solution with the fourth order with respect to the grid size is proved     

Coarse grid correction by aggregation / disaggregation with application in image reconstruction

In algebraic reconstruction of images in computerized tomography we are dealing with rectangular, large, sparse and ill-conditioned linear systems of equations. In this paper we describe a two-grid algorithm for solving such kind of linear systems, which uses Kaczmarz's projection method as relaxation. The correction step is performed with a special “local” aggregation / disaggregation type procedure. In this respect, we have to solve a small sized minimization problem associated to each coarse grid pixel. The information so obtained is then “distributed” to the neighbour fine grid pixels. Some image reconstruction experiments are also presented. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local defect correction with different grid types

For a Poisson problem with a solution having large gradients in (nearly) circular subregions a local defect correction method is considered. The problem on the global domain is discretized on a cartesian grid, whereas the restriction of the problem to a circular subdomain is discretized on a polar grid. The two discretizations are then combined in an iterative way. We show that LDC can be viewed as an iterative method for the Poisson equation on a single composite cartesian‐polar grid. The efficiency of methods is illustrated by numerical examples. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 454–468, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10018     

Domain-decomposition approach to local grid refinement in finite element collocation

Advection-dominated flows occur widely in the transport of groundwater contaminants, the movements of fluids in enhanced oil recovery projects, and many other contexts. In numerical models of such flows, adaptive local grid refinement is a conceptually attractive approach for resolving the sharp fronts or layers that tend to characterize the solutions. However, this approach can be difficult to implement in practice. A domain decomposition method developed by Bramble, Ewing, Pasciak, and Schatz, known as the BEPS method, overcomes many of the difficulties. We demonstrate the applicability of BEPS ideas to finite element collocation on trial spaces of piecewise Hermite cubics. The resulting scheme allows one to refine selected parts of a spatial grid without destroying algebraic efficiencies associated with the original coarse grid. We apply the method to steady-state problems with boundary and interior layers and a time-dependent advection-diffusion problem.     

Preconditioner for a discrete Laplace operator on a condensed grid

In this paper, it is proved that the discrete Laplace operator approximating a Dirichlet boundary value problem for a Poisson equation by a finite element method with piecewise-linear functions on an evenly condensed grid that is topologically equivalent to a rectangular grid (i.e., obtained by shifting the rectangular grid nodes) is equivalent, in the range, to the operator of a 5-point difference scheme on a uniform grid.     

Modified combined grid method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped

A modified combined grid method is proposed for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped. The six-point averaging operator is applied at next-to-the-boundary grid points, while the 18-point averaging operator is used instead of the 26-point one at the remaining grid points. Assuming that the boundary values given on the faces have fourth derivatives satisfying the Hölder condition, the boundary values on the edges are continuous, and their second derivatives obey a matching condition implied by the Laplace equation, the grid solution is proved to converge uniformly with the fourth order with respect to the mesh size.     

A new two grid variational multiscale method for steady‐state natural convection problem

A two‐grid variational multiscale method based on two local Gauss integrations for solving the stationary natural convection problem is presented in this article. A significant feature of the method is that we solve the natural convection problem on a coarse mesh using finite element variational multiscale method based on two local Gauss integrations firstly, and then find a fine grid solution by solving a linearized problem on a fine grid. In the computation, we introduce two local Gauss integrations as a stabilizing term to replace the projection operator without adding other variables. The stability estimates and convergence analysis of the new method are derived. Ample numerical experiments are performed to validate the theoretical predictions and demonstrate the efficiency of the new method. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonnegative splines, in particular of degree five

Summary. We investigate splines from a variational point of view, which have the following properties: (a) they interpolate given data, (b) they stay nonnegative, when the data are positive, (c) for a given integer they minimize the functional for all nonnegative, interpolating . We extend known results for to larger , in particular to and we find general necessary conditions for solutions of this restricted minimization problem. These conditions imply that solutions are splines in an augmented grid. In addition, we find that the solutions are in and consist of piecewise polynomials in with respect to the augmented grid. We find that for general, odd there will be no boundary arcs which means (nontrivial) subintervals in which the spline is identically zero. We show also that the occurrence of a boundary arc in an interval between two neighboring knots prohibits the existence of any further knot in that interval. For we show that between given neighboring interpolation knots, the augmented grid has at most two additional grid points. In the case of two interpolation knots (the local problem) we develop polynomial equations for the additional grid points which can be used directly for numerical computation. For the general (global) problem we propose an algorithm which is based on a Newton iteration for the additional grid points and which uses the local spline data as an initial guess. There are extensions to other types of constraints such as two-sided restrictions, also ones which vary from interval to interval. As an illustration several numerical examples including graphs of splines manufactured by MATLAB- and FORTRAN-programs are given. Received November 16, 1995 / Revised version received February 24, 1997     

Mitigating the curse of dimensionality: sparse grid characteristics method for optimal feedback control and HJB equations

We address finding the semi-global solutions to optimal feedback control and the Hamilton–Jacobi–Bellman (HJB) equation. Using the solution of an HJB equation, a feedback optimal control law can be implemented in real-time with minimum computational load. However, except for systems with two or three state variables, using traditional techniques for numerically finding a semi-global solution to an HJB equation for general nonlinear systems is infeasible due to the curse of dimensionality. Here we present a new computational method for finding feedback optimal control and solving HJB equations which is able to mitigate the curse of dimensionality. We do not discretize the HJB equation directly, instead we introduce a sparse grid in the state space and use the Pontryagin’s maximum principle to derive a set of necessary conditions in the form of a boundary value problem, also known as the characteristic equations, for each grid point. Using this approach, the method is spatially causality free, which enjoys the advantage of perfect parallelism on a sparse grid. Compared with dense grids, a sparse grid has a significantly reduced size which is feasible for systems with relatively high dimensions, such as the 6-D system shown in the examples. Once the solution obtained at each grid point, high-order accurate polynomial interpolation is used to approximate the feedback control at arbitrary points. We prove an upper bound for the approximation error and approximate it numerically. This sparse grid characteristics method is demonstrated with three examples of rigid body attitude control using momentum wheels.     

Multi‐grid for PDEs on complicated domains

A multi‐grid method for solving linear equation systems as they arise from finite element discretisations of elliptic boundary value problems on complicated domains will be presented and analysed. The emphasis is on the robustness of this algorithm with respect to the geometric details in the domain. Robust multi‐grid convergence can be proved for a two‐dimensional model problem.