Computable finite element error bounds for Poisson's equation

Theoretical error bounds of the form are often available for finite element solutions U of elliptic problems. In this form the estimates give the order of convergence of the method but are of little practical value for estimating the size of the error because the magnitudes of the constant K and the theoretical solution u are unknown. An exception occurs in the case of the equation ?²u/?x² + ?²u/?y² + f = 0 in a rectangle where the Ritz–Galerkin finite element solution involves piecewise linears over a regular triangular grid. In this case where α = 1 and Barnhill and Gregory¹ have obtained the theoretical value 0·93√2 for K. In this note calculations are carried out for a variety of problems and the quantity K* = ∥u – U∥_E/h∥f∥_L2 measured and compared with K. The values of K* obtained fit into a well defined pattern from which we conclude that the theoretical constant K is of the correct order of magnitude.     

Accelerating boundary integral equation method using a special‐purpose computer

We present a new solution to accelerate the boundary integral equation method (BIEM). The calculation time of the BIEM is dominated by the evaluation of the layer potential in the boundary integral equation. We performed this task using MDGRAPE‐2, a special‐purpose computer designed for molecular dynamics simulations. MDGRAPE‐2 calculates pairwise interactions among particles (e.g. atoms and ions) using hardwired‐pipeline processors. We combined this hardware with an iterative solver. During the iteration process, MDGRAPE‐2 evaluates the layer potential. The rest of the calculation is performed on a conventional PC connected to MDGRAPE‐2. We applied this solution to the Laplace and Helmholtz equations in three dimensions. Numerical tests showed that BIEM is accelerated by a factor of 10–100. Our rather naive solution has a calculation cost of O(N² × N_iter), where N is the number of unknowns and N_iter is the number of iterations. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical solution of a fourth-order ordinary differential equation

Summary In this paper we have derived numerical methods of orderO(h ⁴) andO(h ⁶) for the solution of a fourth-order ordinary differential equation by finite differences. A method ofO(h ²) was earlier discussed by Usmani and Marsden [6]. Convergence of the fourth-order method is shown. Two examples are computed to show the superiority of our methods.     

A numerical analysis of passive scalar transport in turbulent shear flows

Abstract

The development of the formation and vortex pairing process in a two‐dimensional shear flow and the associated passive scalar (mass concentration or energy) transport process was numerically simulated by using the Vortex‐in‐Cell (VIC) Method combined with the Upwind Finite Difference Method. The visualized temporal distributions of passive scalars resemble the vortex structures and the turbulent passive scalar fluxes showed a definite connection with the occurrence of entrainment during the formation and pairing interaction of large‐scale vortex structures. The profiles of spatial‐averaged passive scalar ø, turbulent passive scalar fluxes, u'ø’ and v'ø’, turbulent diffusivity of mean‐squared scalar fluctuation, v'ø‘ ², mean‐squared turbulent passive scalar fluctuation, √ø‘ ², skewness, and flatness factor of the probability density function of scalar fluctuation ø’ at three different times are calculated. With the lateral dimension scaled by the momentum thickness and the velocity scaled by the velocity difference across the shear layer, these profiles were shown to be self‐preserved. The probability density function of turbulent scalar fluctuation was found to be asymmetric and double‐peaked.     

Alternative analytical solutions of the diffusion (thermal conductivity) equation for an arbitrary initial concentration (temperature) distribution

Alternative analytical solutions of the diffusion (or thermal conductivity) equation are presented, which ensure rapid convergence even for small values of Dt/l ² (αt/l ²), where D is the diffusion coefficient, α is the thermal diffusivity, t is the time, and l is the characteristic size. The solutions possess a general character and are valid for an arbitrary initial distribution of the concentration (temperature).     

A general equation of state for dense fluids

A general equation of state, originally proposed for compressed solids by Parsafar and Mason, has been successfully applied to dense fluids. The equation was tested with experimental data for 13 fluids, including polar, nonpolar, saturated and unsaturated hydrocarbons, strongly hydrogen bonded, and quantum fluids. This equation works well for densities larger than the Boyle density ρ_B [1/ρ_B=T _B d B ₂(T _B)/T], where B₂(T_B) is the second virial coefficient at the Boyle temperature, at whichB ₂=0 and for a wide temperature range, specifically from the triple point to the highest temperature for which the experimental measurements have been reported. The equation is used to predict some important known regularities for dense fluids, like the common bulk modulus and the common compression points, and the Tait-Murnaghan equation. Regarding the common points, the equation of state predicts that such common points are only a low-temperature characteristic of dense fluids as verifed experimentally. It is also found that the temperature dependence of the parameters of the equation of state differs from those given for the compressed solids. Specifically they are given byA ₁ (T)=a ₁+b₁T+c₁T²-d₁ T ln (T).     

Finite elements for an optimal control problem governed by a parabolic equation

A stationary variational formulation of the necessary conditions for optimality is derived for an optimal control problem governed by a parabolic equation and mixed boundary conditions. Then a mixed finite element model with elements in space and time is utilized to solve a simple numerical example whose analytical and finite difference solutions are given elsewhere. Numerical results show that the proposed method with C^° continuity elements constitutes a powerful numerical technique for solution of optimal control problems of distributed parameter systems.     

Finite element solution of a boundary integral equation for mode I embedded three-dimensional fractures

The singular integral equation governing the opening of a mode I embedded three-dimensional fracture in an infinite solid was solved by applying the finite element method. The strategy is to formulate the equation into weak form, and to transfer the differentiation from the singular term, 1/r, in the equation to the test function. A numerical algorithm was thus developed. The numerical solutions for circular and elliptical fractures under the action of polynomial pressure distributions were compared with the analytical solutions by Green and Sneddon,¹² Irwin,¹³ Shah and Kobayashi¹⁴ and Nishioka and Atluri.¹⁶ The results have demonstrated that the numerical method reported is accurate and efficient.     

The radial basis integral equation method for solving the Helmholtz equation

A meshless method for the solution of Helmholtz equation has been developed by using the radial basis integral equation method (RBIEM). The derivation of the integral equation used in the RBIEM is based on the fundamental solution of the Helmholtz equation, therefore domain integrals are not encountered in the method. The method exploits the advantage of placing the source points always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green’s identities and the remaining equations are the derivatives of the first equation with respect to space coordinates. Radial basis function (RBF) interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). The accuracy and robustness of the method has been tested on some analytical solutions of the problem. Two different RBFs have been used, namely augmented thin plate spline (ATPS) in 2D and f(R)=⁴Rln(R) augmented by a second order polynomial. The latter has been found to produce more accurate results.     

Exterior stable domain segmentation integral equation method for scattering problems

This paper is concerned with the development of an exterior domain segmentation method for the solution of two- or three-dimensional time-harmonic scattering problems in acoustic media. This method, based on a variational localized, symmetric, boundary integral equation formulation leads, upon discretization, to a sparse system of algebraic equations whose solution requires only O(N) multiplications, where N is the number of unknown nodal pressures on the scatterer surface. The new procedure is analogous to the one developed recently¹ except that in the present formulation we avoid completely the use of the hypersingular operator, thereby reducing the computational complexity. Numerical experiments for a rigid circular cylindrical scatterer subjected to a plane incident wave serve to assess its accuracy for normalized wave numbers, ka, ranging from 0 to 30, both directly on the scatterer and in the far field, and to confirm that, contrary to standard boundary integral equation formulations, the present procedure is valid for critical frequencies.     

A fourth‐order compact scheme for the Helmholtz equation: Alpha‐interpolation of FEM and FDM stencils

We propose a fourth‐order compact scheme on structured meshes for the Helmholtz equation given by R(φ):=f( x )+Δφ+ξ²φ=0. The scheme consists of taking the alpha‐interpolation of the Galerkin finite element method and the classical central finite difference method. In 1D, this scheme is identical to the alpha‐interpolation method (J. Comput. Appl. Math. 1982; 8 (1):15–19) and in 2D making the choice α=0.5 we recover the generalized fourth‐order compact Padé approximation (J. Comput. Phys. 1995; 119 :252–270; Comput. Meth. Appl. Mech. Engrg 1998; 163 :343–358) (therein using the parameter γ=2). We follow (SIAM Rev. 2000; 42 (3):451–484; Comput. Meth. Appl. Mech. Engrg 1995; 128 :325–359) for the analysis of this scheme and its performance on square meshes is compared with that of the quasi‐stabilized FEM (Comput. Meth. Appl. Mech. Engrg 1995; 128 :325–359). In particular, we show that the relative phase error of the numerical solution and the local truncation error of this scheme for plane wave solutions diminish at the rate O((ξ?)⁴), where ξ, ? represent the wavenumber and the mesh size, respectively. An expression for the parameter α is given that minimizes the maximum relative phase error in a sense that will be explained in Section 4.5. Convergence studies of the error in the L² norm, the H¹ semi‐norm and the l^∞ Euclidean norm are done and the pollution effect is found to be small. Copyright © 2010 John Wiley & Sons, Ltd.     

Solution of the compressible flow equations

A technique is described for solving the compressible flow equations in subsonic flow. The general quasi-linear equation ?.g?v = 0 is considered with g a function of ?v. ?v, and iterations of the form ?.g_n?v_n+1 = 0 are analysed, where g₀ is suitably chosen and g_n defined from v_n for n≥1. This approach is applied to the compressible flow equations in terms of a velocity potential ø: monotonic convergence is predicted and at each iteration the error is multiplied by a factor less than the square of the greatest Mach number in the solution. Reliable convergence has been obtained in practice solving the linear equation for ø_n+1 by a finite difference method. The alternative of working in terms of the stream function ψ is discussed, and also discretization by the finite element method.     

A posteriori error estimates of mixed methods for miscible displacement problems

The miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is an elliptic equation of the pressure and the other is a parabolic equation of the concentration of one of the fluids. Since the pressure appears in the concentration only through its velocity field, we choose a mixed finite element method to approximate the pressure equation and for the concentration we use the standard Galerkin method. We shall obtain an explicit a posteriori error estimator in L²(L²) for the semi‐discrete scheme applied to the non‐linear coupled system. Copyright © 2007 John Wiley & Sons, Ltd.     

High order difference schemes for the wave equation

ADI difference schemes for the solution of the wave equation with variable coefficients are given. Mckee¹ has given a formula of order O(k² + h⁴), while Ciment and Leventhal² have given a formula of order O(k⁴ + h⁴). Both the formulas are conditionally stable with the condition λ√c λ √3?1. In this paper we have derived uncoditionally stable ADI schemes of orders O(k² + h²) and O(k² + h⁴). These schemes allow large time steps and the computational effort is thus reduced. Numerical experiments suggest the usefulness of our formulas.     

The measurement of mobilities and doping concentrations in field-effect-controlled thin semiconductor films

A method is proposed for determining the functional relationship between the carrier mobility and the electrostatic potential in uniform n- or p-type thin film semiconductors from experimental Hall measurements. Use of a symmetrical Hall effect MISIM thin film measuring structure yields tractable solutions of Poisson's equation and should help circumvent problems of interpretation that might otherwise arise when using an asymmetric MIS configuration.In the analytic method demonstrated for an n-type semiconductor with field-plate-induced accumulation layers, two expressions are obtained relating donor density N_d and carrier drift mobility μ(?) in terms of measurable Hall effect parameters as a function of normalized electrostatic potential ø and normalized film thickness 2λ_d. A given example using a plausible functional relationship for μ(ø) indicates that considerable deviations between μ(ø) and the Hall mobility <μ_H(ø_s)> can exist. A numerical procedure for solving the integral equations relating μ(ø) and <μ_H(ø_s)> is presented.     

Exact finite element solutions to the Helmholtz equation

For one-, two- and three-dimensional co-ordinate systems finite element matrices for the wave or Helmholtz equation are used to produce a single difference equation holding at any point of a regular mesh. Under homogeneous Dirichlet or Neumann boundary conditions, these equations are solved exactly. The eigenfunctions are the discrete form of sine or cosine functions and the eigenvalues are shown to be in error by a term of + O(h²ⁿ) where n is the order of the polynomial approximation of the wave function. The solutions provide the means of testing computer programs against the exact solutions and allow comparison with other difference schemes.     

Algorithms for direct-access gaussian solution of structural stiffness matrix equation

Based on direct-access programming, algorithms have been developed for the generation, and solution by Gaussian elimination of the structural stiffness matrix equation resulted from application of the finite element method in engineering analyses. A large disk storage is used to store the rows of the stiffness matrix as directly accessible records. The developed algorithm BAND2R requires only 2N_b in-core words in implementing the Gaussian elimination where N_b is the semi-band width of the stiffness matrix. Algorithms BANDSQ and BDSQMX are presented which require N²_b in-core words but minimize the number of retrieving and restoring the direct-access records during the Gaussian elimination. BANDSQ has the direct-access feature in both the elimination and backward-substitution steps whereas BDSQMX has the direct-access feature only in the backward-substitution step of the Gaussian elimination. Illustrative applications of the developed algorithms are given and the computer core and time requirements for BAND2R, BANDSQ and BDSQMX are compared to those for the conventional Gaussian elimination of using sequential, in-core storages. Methods for reducing the semi-band width N_b of the structural stiffness matrix are also discussed.     

A relation between the solutions of the half-space Dirichlet problems for Helmholtz's equation in ℝ n and Laplace's equation in ℝ n+1

Summary Multiple Fourier transforms are used to derive the solutions of the half-space Dirichlet problems for Helmholtz's equation in ℝ ⁿ and Laplace's equation in ℝ ⁿ⁺¹ and to exhibit the relation between the two solutions.     

Wide-range equation of state for water and steam: Method of construction

A wide-range equation of state for water and steam in analytical form is developed. The equation is valid for moderate and high pressures up to 4 × 10¹² Pa, in particular, explosive and static compression in a wide range of densities. An equation is derived which enables one to calculate the Grueneisen coefficient dependent on specific volume and temperature. A method is suggested which enables one to use the experimental data for the dependences of heat capacity and of isochoric coefficient of temperature pressure increase on specific volume and temperature for calculating the Grueneisen coefficient and internal heat energy.     

Transformation of continuous‐time state equations to discrete‐time state equations

Abstract

A method is proposed for transforming a continuous‐time state equation, x(t)=Ax(t)+Bu(t), to a discrete‐time state equation, x[(k+1)τ]=ø(τ)x(kτ)+B(τ)u(kτ). It is based on expanding the matrix exponential exp (Aτ) into a shifted matrix Chebyshev series. An example is given to demonstrate the superiority of the method over other methods.