Optimization of computational grids

We consider the problem of mesh grading or gridr-point redistribution using optimization techniques. Some representate forms of the objective functions are considered and the assocated scaling parameterization and nonlinear programming algorithms examined. Numerical results for representative test problems involving two-and three-dimensional grids are presented.     

Optimal adaptive grids of least-squares finite element methods in two spatial dimensions

This article concerns a procedure to generate optimal adaptive grids for convection dominated problems in two spatial dimensions based on least-squares finite element approximations. The procedure extends a one dimensional equidistribution principle which minimizes the interpolation error in some norms. The idea is to select two directions which can reflect the physics of the problems and then apply the one dimensional equidistribution principle to the chosen directions. Model problems considered are the two dimensional convection-diffusion problems where boundary and interior layers occur. Numerical results of model problems illustrating the efficiency of the proposed scheme are presented. In addition, to avoid skewed mesh in the optimal grids generated by the algorithm, an unstructured local mesh smoothing will be considered in the least-squares approximations. Comparisons with the Gakerkin finite element method will also be provided.     

The grid refinement technique

The grid refinement technique is a discrete iterative procedure which generates sequences of approximate solutions to two-point variational problems defined on continuous fields. The approximate solutions are obtained by calculating the optimal path through discrete grids constructed throughout the continuous field. Convergence of the series of discrete approximating paths is obtained by successively decreasing the mesh size of the grids.     

Hermitian splines as basis functions of the finite-element method for plotting stress trajectories

We propose a finite-element approach with the use of Hermitian splines for reducing the problem of plotting of stress trajectories for finding two potential functions that satisfy conditions of interpolation of first derivatives at nodes of a regular mesh. The efficiency of this approach is demonstrated on test problems, and features of computational relations are determined.     

Optimal multilevel methods for graded bisection grids

We design and analyze optimal additive and multiplicative multilevel methods for solving H ¹ problems on graded grids obtained by bisection. We deal with economical local smoothers: after a global smoothing in the finest mesh, local smoothing for each added node during the refinement needs to be performed only for three vertices - the new vertex and its two parent vertices. We show that our methods lead to optimal complexity for any dimensions and polynomial degree. The theory hinges on a new decomposition of bisection grids in any dimension, which is of independent interest and yields a corresponding decomposition of spaces. We use the latter to bridge the gap between graded and quasi-uniform grids, for which the multilevel theory is well-established.     

Efficient iterative solvers for time-harmonic Maxwell equations using domain decomposition and algebraic multigrid

Paper presents a set of parallel iterative solvers and preconditioners for the efficient solution of systems of linear equations arising in the high order finite-element approximations of boundary value problems for 3-D time-harmonic Maxwell equations on unstructured tetrahedral grids. Balancing geometric domain decomposition techniques combined with algebraic multigrid approach and coarse-grid correction using hierarchic basis functions are exploited to achieve high performance of the solvers and small memory load on the supercomputers with shared and distributed memory. Testing results for model and real-life problems show the efficiency and scalability of the presented algorithms.     

Finite-element methods for singularly perturbed high-order elliptic two-point boundary value problems. I: reaction-diffusion-type problems

We consider singularly perturbed high-order elliptic two-pointboundary value problems of reaction-diffusion type. It is shownthat, on an equidistant mesh, polynomial schemes cannot achievea high order of convergence that is uniform in the perturbationparameter. Piecewise polynomial Galerkin finite-element methodsare then constructed on a Shishkin mesh. Almost optimal convergenceresults, which are uniform in the perturbation parameter, areobtained in various norms. Numerical results are presented fora fourth-order problem. e-mail address: stynes{at}bureau.ucc.ie.     

The hierarchical preconditioning on unstructured three-dimensional grids with locally refined regions

This paper presents two hierarchically preconditioned methods for the fast solution of mesh equations that approximate three-dimensional-elliptic boundary value problems on quasiuniform triangulations above all aiming at the numerical investigation of the previously suggested algorithms. Furthermore, improving the practical applicability of the methods unstructured three-dimensional grids possessing locally refined regions are considered. Based on the fictitious space approach, the original problem can be adaptively embedded into an auxiliary one in which hanging nodes occur. We implemented the corresponding Yserentant preconditioned conjugate gradient method as well as the BPX-preconditioned cg-iteration having nearly optimal computational costs. Several numerical examples demonstrate the efficiency of the artificially constructed hierarchical methods.     

Recovering Mesh Geometry from a Stiffness Matrix

We introduce the following class of mesh recovery problems: Given a stiffness matrix A and a PDE, construct a mesh M such that the finite-element formulation of the PDE over M is A. We show, under certain assumptions, that it is possible to reconstruct the original mesh for the special case of the Laplace operator discretized on an unstructured mesh of triangular elements with linear basis functions. The reconstruction is achieved through a series of techniques from graph theory and numerical analysis, some of which are new and can find application in other scientific areas. Finally, we discuss extensions to other operators and some open questions related to this class of problems.     

A conforming finite element method for overlapping and nonmatching grids

In this paper we propose a finite element method for nonmatching overlapping grids based on the partition of unity. Both overlapping and nonoverlapping cases are considered. We prove that the new method admits an optimal convergence rate. The error bounds are in terms of local mesh sizes and they depend on neither the overlapping size of the subdomains nor the ratio of the mesh sizes from different subdomains. Our results are valid for multiple subdomains and any spatial dimensions.

     

Node-pair finite volume/finite element schemes for the Euler equation in cylindrical and spherical coordinates

A numerical scheme is presented for the solution of the compressible Euler equations in both cylindrical and spherical coordinates. The unstructured grid solver is based on a mixed finite volume/finite element approach. Equivalence conditions linking the node-centered finite volume and the linear Lagrangian finite element scheme over unstructured grids are reported and used to devise a common framework for solving the discrete Euler equations in both the cylindrical and the spherical reference systems. Numerical simulations are presented for the explosion and implosion problems with spherical symmetry, which are solved in both the axial–radial cylindrical coordinates and the radial–azimuthal spherical coordinates. Numerical results are found to be in good agreement with one-dimensional simulations over a fine mesh.     

Adaptive least-squares finite element approximations to transonic-flow problems

This work concerns solutions of compressible potential flow problems based on weighted least-squares finite element approximations. The model problem considered is that of the potential flow past a circular cylinder. To capture the transonic flow region, an adaptive algorithm based on mesh redistribution with local mesh refinement and smoothing is developed for suitably weighted least-squares approximations. Numerical results for the model problem are given for the transonic case with shocks. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On solution strategies to Saint-Venant problem

Different solution strategies to the relaxed Saint-Venant problem are presented and comparatively discussed from a mechanical and computational point of view. Three approaches are considered; namely, the displacement approach, the mixed approach, and the modified potential stress approach. The different solution strategies lead to the formulation of two-dimensional Neumann and Dirichlet boundary-value problems. Several solution strategies are discussed in general, namely, the series approach, the reformulation of the boundary-value problems for the Laplace's equations as integral boundary equations, and the finite-element approach. In particular, the signatures of the finite-element weak solutions—the computational costs, the convergence, the accuracy—are discussed considering elastic cylinders whose cross sections are represented by piece-wise smooth domains.     

Construction and study of high-order accurate schemes for solving the one-dimensional heat equation

The method of undetermined coefficients on multipoint stencils with two time levels was used to construct compact difference schemes of O(τ³, h ⁶) accuracy intended for solving boundary value problems for the one-dimensional heat equation. The schemes were examined for von Neumann stability, and numerical experiments were conducted on a sequence of grids with mesh sizes tending to zero. One of the schemes was proved to be absolutely stable. It was shown that, for smooth solutions, the high order of convergence of the numerical solution agrees with the order of accuracy; moreover, solutions accurate up to ～10^?12 are obtained on grids with spatial mesh sizes of ～10^?2. The formulas for the schemes are rather simple and easy to implement on a computer.     

Nested dyadic grids associated with Legendre–Gauss–Lobatto grids

Legendre–Gauss–Lobatto (LGL) grids play a pivotal role in nodal spectral methods for the numerical solution of partial differential equations. They not only provide efficient high-order quadrature rules, but give also rise to norm equivalences that could eventually lead to efficient preconditioning techniques in high-order methods. Unfortunately, a serious obstruction to fully exploiting the potential of such concepts is the fact that LGL grids of different degree are not nested. This affects, on the one hand, the choice and analysis of suitable auxiliary spaces, when applying the auxiliary space method as a principal preconditioning paradigm, and, on the other hand, the efficient solution of the auxiliary problems. As a central remedy, we consider certain nested hierarchies of dyadic grids of locally comparable mesh size, that are in a certain sense properly associated with the LGL grids. Their actual suitability requires a subtle analysis of such grids which, in turn, relies on a number of refined properties of LGL grids. The central objective of this paper is to derive the main properties of the associated dyadic grids needed for preconditioning the systems arising from \(hp\)- or even spectral (conforming or Discontinuous Galerkin type) discretizations for second order elliptic problems in a way that is fully robust with respect to varying polynomial degrees. To establish these properties requires revisiting some refined properties of LGL grids and their relatives.     

Optimal control in coefficients of elliptic equations with state constraints

This paper is concerned with state constrained optimal control problems of elliptic equations, the control being a coefficient of the partial differential equation. Existence of an optimal control is proved and optimality conditions are derived. We perform finite-element approximations of optimal control problems and state some convergence results: we prove convergence of optimal controls and states as well as convergence of Lagrange multipliers.This research was partially supported by the Dirección General de Investigación Científica y Técnica (Madrid).     

An interface-fitted adaptive mesh method for elliptic problems and its application in free interface problems with surface tension

This work presents a novel two-dimensional interface-fitted adaptive mesh method to solve elliptic problems of jump conditions across the interface, and its application in free interface problems with surface tension. The interface-fitted mesh is achieved by two operations: (i) the projection of mesh nodes onto the interface and (ii) the insertion of mesh nodes right on the interface. The interface-fitting technique is combined with an existing adaptive mesh approach which uses addition/subtraction and displacement of mesh nodes. We develop a simple piecewise linear finite element method built on this interface-fitted mesh and prove its almost optimal convergence for elliptic problems with jump conditions across the interface. Applications to two free interface problems, a sheared drop in Stokes flow and the growth of a solid tumor, are presented. In these applications, the interface surface tension serves as the jump condition or the Dirichlet boundary condition of the pressure, and the pressure is solved with the interface-fitted finite element method developed in this work. In this study, a level-set function is used to capture the evolution of the interface and provide the interface location for the interface fitting.     

External finite-element approximations of eigenfunctions in the case of multiple eigenvalues

The paper deals with the finite-element analysis of second-order elliptic eigenvalue problems when the approximate domains Ω_h are not subdomains of the original domain . The considerations are restricted to piecewise linear approximations. Special attention is devoted to the convergence of approximate eigenfunctions in the case of multiple exact eigenvalues. As yet the approximate solutions have been compared with linear combinations of exact eigenfunctions with coefficients depending on the mesh parameter h. We avoid this disadvantage.     

Multiple-interval pseudospectral approximation for nonlinear optimal control problems with time-varying delays

A multiple-interval pseudospectral scheme is developed for solving nonlinear optimal control problems with time-varying delays, which employs collocation at the shifted flipped Jacobi-Gauss–Radau points. The new pseudospectral scheme has the following distinctive features/abilities: (i) it can directly and flexibly solve nonlinear optimal control problems with time-varying delays without the tedious quasilinearization procedure and the uniform mesh restriction on time domain decomposition, and (ii) it provides a smart approach to compute the values of state delay efficiently and stably, and a unified framework for solving standard and delay optimal control problems. Numerical results on benchmark delay optimal control problems including challenging practical engineering problems demonstrate that the proposed pseudospectral scheme is highly accurate, efficient and flexible.     

Grid approximation of a parabolic convection-diffusion equation on a priori adapted grids: ε-uniformly convergent schemes

The boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered. A finite difference scheme on a priori (sequentially) adapted grids is constructed and its convergence is examined. The construction of the scheme on a priori adapted grids is based on a majorant of the singular component of the grid solution that makes it possible to a priori find a subdomain in which the grid solution should be further refined given the perturbation parameter ε, the size of the uniform mesh in x, the desired accuracy of the grid solution, and the prescribed number of iterations K used to refine the solution. In the subdomains where the solution is refined, the grid problems are solved on uniform grids. The error of the solution thus constructed weakly depends on ε. The scheme converges almost ε-uniformly; namely, it converges under the condition N ^?1 = o(ε^v), where v = v(K) can be chosen arbitrarily small when K is sufficiently large. If a piecewise uniform grid is used instead of a uniform one at the final Kth iteration, the difference scheme converges ε-uniformly. For this piecewise uniform grid, the ratio of the mesh sizes in x on the parts of the mesh with a constant size (outside the boundary layer and inside it) is considerably less than that for the known ε-uniformly convergent schemes on piecewise uniform grids.