Showing papers on "Discretization published in 1984"

Galerkin Finite Element Methods for Parabolic Problems

Abstract: The Standard Galerkin Method.- Methods Based on More General Approximations of the Elliptic Problem.- Nonsmooth Data Error Estimates.- More General Parabolic Equations.- Negative Norm Estimates and Superconvergence.- Maximum-Norm Estimates and Analytic Semigroups.- Single Step Fully Discrete Schemes for the Homogeneous Equation.- Single Step Fully Discrete Schemes for the Inhomogeneous Equation.- Single Step Methods and Rational Approximations of Semigroups.- Multistep Backward Difference Methods.- Incomplete Iterative Solution of the Algebraic Systems at the Time Levels.- The Discontinuous Galerkin Time Stepping Method.- A Nonlinear Problem.- Semilinear Parabolic Equations.- The Method of Lumped Masses.- The H1 and H?1 Methods.- A Mixed Method.- A Singular Problem.- Problems in Polygonal Domains.- Time Discretization by Laplace Transformation and Quadrature.

A Taylor–Galerkin method for convective transport problems

Abstract: A method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions. To produce accurate temporal differencing, the method employs forward-time Taylor series expansions including time derivatives of second- and third-order which are evaluated from the governing partial differential equation. This yields a generalized time-discretized equation which is successively discretized in space by means of the standard Bubnov–Galerkin finite element method. The technique is illustrated first in one space dimension. With linear elements and Euler, leap-frog and Crank–Nicolson time stepping, several interesting relations with standard Galerkin and recently developed Petrov–Galerkin methods emerge and the new Taylor–Galerkin schemes are found to exhibit particularly high phase-accuracy with minimal numerical damping. The method is successively extended to deal with variable coefficient problems and multi-dimensional situations.

Finite element solution of diffusion problems with irregular data

Abstract: Diffusion problems occuring in practice often involve irregularities in the initial or boundary data resulting in a local break-down of the solution's regularity. This may drastically reduce the accuracy of discretization schemes over the whole interval of integration, unless certain precautions are taken. The diagonal Pade schemes of order 2μ, combined with a standard finite element discretization, usually require an unnatural step size restriction in order to achieve even locally optimal accuracy. It is shown here that this restriction can be avoided by means of a sample damping procedure which preserves the order of the discretization and, in the case μ=1, does not increase the costs.

The solution of non‐linear hyperbolic equation systems by the finite element method

Abstract: A finite-element method for the solution of nonlinear hyperbolic systems of equations, such as those encountered in non-self-adjoint problems of transient phenomena in convection-diffusion or in the mixed representation of wave problems, is developed and demonstrated. The problem is rewritten in moving coordinates and reinterpolated to the original mesh by a Taylor expansion prior to a standard Galerkin spatial discretization, and it is shown that this procedure is equivalent to the time-discretization approach of Donea (1984). Numerical results for sample problems are presented graphically, including such shallow-water problems as the breaking of a dam, the shoaling of a wave, and the outflow of a river; compressible flows such as the isothermal flow in a nozzle and the Riemann shock-tube problem; and the two-dimensional scalar-advection, nonlinear-shallow-water, and Euler equations.

Averaged Multivalued Solutions for Scalar Conservation Laws

Abstract: A time discretization is introduced for scalar conservation laws, which consists in averaging (in an appropriate sense) the generally multivalued solution given by the classical method of characteristics. Convergence toward the physical solutions satisfying the entropy condition is proved. Several numerical schemes are deduced after a full discretization, either with respect to the space variable (various known schemes are then recognized), or with respect to the phase variable (which leads to a space grid free scheme). Generalizations are considered toward systems of conservation laws and bidimensional scalar conservation laws.

Approximate solutions of the Bellman equation of deterministic control theory

Abstract: We consider an infinite horizon discounted optimal control problem and its time discretized approximation, and study the rate of convergence of the approximate solutions to the value function of the original problem. In particular we prove the rate is of order 1 as the discretization step tends to zero, provided a semiconcavity assumption is satisfied. We also characterize the limit of the optimal controls for the approximate problems within the framework of the theory of relaxed controls.

Methods for the numerical solution of the nonlinear Schroedinger equation

Abstract: Optimal L2 rates of convergence are established for several fully-discrete schemes for the numerical solution of the nonlinear Schroedinger equation. Both finite differences and finite elements are considered for the discretization in space, while the integration in time is treated either by the leap-frog technique or by a modified Crank-Nicolson procedure, which generalizes the one suggested by Delfour, Fortin and Payne and possesses two useful conserved quantities.

Three-Dimensional Unsteady Euler Equations Solution Using Flux Vector Splitting.

Abstract: A method for numerically solving the three-dimensional unsteady Euler equations using flux vector splitting is developed. The equations are cast in curvilinear coordinates and a finite volume discretization is used. An explicit upwind second-order predictor-corrector scheme is used to solve the discretized equations. The scheme is stable for a CFL number of 2 and local time stepping is used to accelerate convergence for steady-state problems. Characteristic variable boundary conditions are developed and used in the far-field and at surfaces. No additional dissipation terms are included in the scheme. Numerical results are compared with results from an existing three-dimensional Euler code and experimental data.

Boundary integral equation method for linear porous‐elasticity with applications to soil consolidation

Abstract: For physical phenomena governed by the Biot model of porous-elasticity, a reciprocal relation, similar to the Betti's recoprocal theorem in elasticity, is constructed in Laplace transformed space. Integrating the reciprocal relation enables one to formulate boundary integral equations. The fundamental kernels for the integral equations are solved in closed forms for the case of isotropic material. Numerical implementation of two-dimensional problems includes finite element ideas of discretization and polynomial interpolation, and numerical inversion of a Laplace transform. Practical applications of the method are found in consolidation problems in soils which contain compressible as well as incompressible pore fluids. Also, as a numerical experiment, consolidation of partially saturated soil is simulated and interesting phenomena are observed. The currently developed boundary integral equation method (BIEM) for porous-elasticity may be viewed as an efficient and accurate alternative of existing finite element and finite difference methods. For linear consolidation problems, application of BIEM is always preferred to the other numerical methods whenever possible.

Imbricate continuum and its variational derivation

Abstract: The one-dimensional imbricate nonlocal continuum, which was developed in another paper in order to model strain-softening within zones of finite size, is extended here to two or three dimensions. The continuum represents a limit of a system of imbricated (overlapping) elements that have a fixed size and a diminishing cross section as the mesh is refined. The proper variational method for the imbricate continuum is developed, and the continuum equations of motion are derived from the principle of virtual work. They are of difference-differential type and involve not only strain averaging but also stress gradient averaging for the so-called broad-range stresses characterizing the forces within the characteristic volume of heterogeneous material. The gradient averaging may be defined by a difference operator, or an averaging integral, or by least-square fitting of a homogeneous strain field. A differential approximation with higher order displacement derivatives is also shown. The theory implies a boundary layer which requires special treatment. The blunt crack band model, previously used in finite element analysis of progressive fracturing, is extended by the present theory into the range of mesh sizes much smaller than the characteristic width of the crack band front. Thus, the crack band model is made part of a convergent discretization scheme. The nonlocal continuum aspects are captured by an imbricated arrangement of finite elements, which are of the usual type.

Fast-Fourier-Transform Method for Calculation of SAR Distributions in Finely Discretized Inhomogeneous Models of Biological Bodies

Abstract: The paper describes a novel iterative approach for calculations of specific absorption rate (SAR) distributions in arbitrary, lossy, dielectric bodies. To date, the method has been used for 2-D problems where its accuracy has been confirmed by comparison with the analytic solutions for homogeneous and layered, circular, cylindrical bodies. With computation times that are proportional to N log/sub 2/N rather than N/sup 2/ to N/sup 3/ for the method of moments, the present approach should be extendable to 3-D bodies with N= 10/sup 4/to 10/sup 5/ cells allowing, thereby, details of SAR distributions that are needed for EM hyperthermia, as well as for assessing biological effects.

A Review of the Theory

Abstract: The modern theory of boundary integral equations began with Fredholm [1], who established the existence of solutions on the basis of his limiting discretisation procedure. It was not envisaged by Fredholm or his immediate successors that solutions could actually be constructed in this way. However the advent of fast digital computers, some 50 years later, opened up the possibility of implementing the discretisation process arithmetically and so enabled numerical solutions of tolerable accuracy to be attempted. This possibility in turn gave a considerable impetus to the development of new and improved boundary integral formulations. In 1962, Hess and Smith [2, 3] formulated a Fredholm integral equation of the second kind for the distribution of simple sources over a surface of revolution. By solving this equation numerically, they were able to compute the perturbation of a uniform potential flow by the surface. In 1963, Jaswon and Ponter [4] threw the torsion problem on to the boundary by formulating an integral equation of the second kind for the warping function, which was solved numerically as a means of computing the torsional rigidity and boundary shear stress for cross-sections inaccessible to other methods of attack. This was one of the first published papers which effectively exploited Green’s formula on the boundary, by emphasising its role as a functional relation between the boundary values and normal derivatives of an arbitrary harmonic function. Also in 1963, Jaswon [5] formulated the electrostatic capacitance problem in terms of a Fredholm integral equation of the first kind for the charge distribution, a formulation which had been noted and discarded by Volterra [6] because of apparent difficulties with the two-dimensional theory.

Local Defect Correction Method and Domain Decomposition Techniques

Abstract: For elliptic problems a local defect correction method is described. A basic (global) discretization is improved by a local discretization defined in a subdomain. The convergence rate of the local defect correction iteration is proved to be proportional to a certain positive power of the step size. The accuracy of the converged solution can be described. Numerical examples confirm the theoretical results. We discuss multi-grid iterations converging to the same solution.

Analysis of Hybrid Field Problems by the Method of Lines with Nonequidistant Discretization

Abstract: The method of lines, which has been proved to be very efficient for calculating the characteristics of one-dimensional and two-dimensional planar microwave structures, is extended to nonequidistant discretization. By means of an intermediate transformation it is possible to maintain all essential transformation properties that are given in the case of equidistant discretization. The flexibltity of the method of lines is increased substantially. As a consequence, the accuracy is improved with reduced computational effort.

Time Finite Element Discretization of Hamilton's Law of Varying Action

Abstract: Hamilton's Law of Varying Action is used as a variational source for the derivation of finite element discretization procedure in the time domain. Three different versions of the proposed algorithms are presented and verified for accuracy and stability. [The first one is the high-precision, finite time element, analogous to the standard finite elements, with cutoff frequency; the second version is the step by step, one-time element from which the unconditionally stable, with slightly altered accuracy, third algorithm is derived. ] The new operator, connected with the proposed algorithms, bears attractive properties of much greater accuracy than other existing stable methods, and easy computer implementation. Thus, the work herein shows that the reservations expressed against the use of finite elements in time domain seem unjustified.

Finite element mesh generation

Abstract: The capabilities of a geometric modeller are extended towards finite element analysis by a mesh generator which extracts all its geometric and topological information from the model. A coarse mesh is created and subsequently refined to a suitable finite element mesh, which accomodates material properties, loadcase and analysis requirements. The mesh may be optimized by adaptive refinement, ie according to estimates of the discretization errors. A survey of research and development in geometric modelling and finite element analysis is presented, then an implementation of a mesh generator for 3D curvilinear and solid objects is described in detail.

An Interface Tracking Algorithm for the Porous Medium Equation.

Abstract: We study the convergence of a finite difference scheme for the Cauchy problem for the porous medium equation u, = («m)AV, m > 1. The scheme exhibits the following two features. The first is that it employs a discretization of the known interface condition for the propagation of the support of the solution. We thus generate approximate interfaces as well as an approximate solution. The second feature is that it contains a vanishing viscosity term. This term permits an estimate of the form ||(""i_1)a.vIIi.r < c/t. We prove that both the approximate solution and the approximate interfaces converge to the correct ones. Finally error bounds for both solution and free boundaries are proved in terms of the mesh parameters. 1. Introduction. In this paper we derive and analyze a finite difference scheme for computing both the solution and the interfaces for the porous medium equation in one space dimension. We demonstrate that the approximate solutions and the approximate interface curves converge to the correct ones, and we obtain L00 bounds for the error in terms of the mesh parameter. Consider the laminar flow of a polytropic fluid of density (x, t) -* u(x, t) in a porous medium which is assumed to occupy the whole space, and suppose that at time t = 0 the fluid is contained in the slab £,(0) < x < fr(0). The phenomenon can be modeled by

Parallel S.O.R. iterative methods

Abstract: New explicit group S.O.R. methods suitable for use on an asynchronous MIMD computer are presented for the numerical solution of the sparse linear systems derived from the discretization of two-dimensional, second-order, elliptic boundary value problems. A comparison with existing implicit line S.O.R. schemes for the Dirichlet model problem shows the new schemes to be superior (Barlow and Evans, 1982).

Numerical solution of partial differential equations with variable coefficients with an operational approach to the tau method

Abstract: We discuss the numerical solution of linear partial differential equations with variable coefficients by means of an operational approach to Ortiz' recursive formulation of the Tau method. We discuss a procedure which makes it possible to determine the coefficients of a bivariate Tau approximant by means of a reduced set of matrix operations. It involves no discretization of the variables, approximate quadratures or the use of special trial functions. Error surfaces exhibit a remarkable equioscillatory behaviour.

Finite-Difference Solutions of Convection-Diffusion Problems in Irregular Domains, Using a Nonorthogonal Coordinate Transformation

Abstract: A solution methodology has been developed for convection-diffusion problems in which one boundary of the solution domain does not lie along a coordinate line. A nonorthogonal, algebraic coordinate transformation is used which yields a rectangular solution domain. This transformation avoids the task of numerically generating boundary-fitted coordinates. The discretized conservation equations are derived on a control-volume basis. These equations contain pseudodiffusion terms that result from the nonorthogonal nature of the transformation. The entire discretization procedure is documented in detail. Although it is not an essential feature of the method, the discretized equations and their solutions are tied in with the well-documented practices of the Patankar solution scheme for orthogonal systems. Application of the methodology is illustrated by two numerical examples.

Toward a stable marching‐on‐in‐time method for two‐dimensional transient electromagnetic scattering problems

Abstract: The transient scattering of two-dimensional electromagnetic fields by an obstacle of finite extent is investigated with the aid of the time domain integral equation technique. In solving such equations with the marching-on-in-time method, numerical instabilities form a major problem. These instabilities can be attributed to errors in the discretization of the source type integrals that occur in the equations. In this paper, we formulate two so-called stability criteria for such a discretization that, if they are met, guarantee that the instability can be controlled by reducing the discretization step. With the aid of these criteria, we analyze the solution of two two-dimensional electromagnetic scattering problems, namely the scattering of a pulsed plane wave by a perfectly conducting and an inhomogeneous, lossy dielectric cylinder. Numerical results are presented and discussed.