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The stability ofL 2-projection into many standard finite element spaces as a map intoL q , 1?q??, is demonstrated. TheL 2-projection ofu?L q is shown to be a quasi-optimal approximation ofu inL q .

The method of Gordon for the solution of the Schrodinger equation for elastic scattering is reformulated. Ordinary Rayleigh-Schrodinger perturbation theory is used to obtain the solution in a succession of intervals of the independent (radial) variable. A criterion for the automatic selection of interval sizes for a requisite accuracy of the phase shift is developed. The perturbation technique (carried to first order, taking the zero-order potential to be constant) is tested against the highly efficient Numerov direct-integration method on the Lennard-Jones (12,6) potential. It is found that, under the restrictions imposed on the perturbation method, the Numerov procedure is almost always more efficient, except for partial waves of low angular momentum.

Basic concepts of asymptotic theory for discretization methods have been difined so that convergence is always equivalent to consistency and stanility. The theory is then applied to a class of discretization methods for the initial value problems of ordinary differential equations, the so-called weakly nonlinear methods.

Dual variational principles for steady state wave propagation in three dimensional thermoviscoelastic media are presented. The first one, for the equations of motion, involves only the complex displacement function. The second principle is for the energy equation. These principles are specialized to the case of one dimension. A one-dimensional example, that of wave propagation in a thermoviscoelastic rod insulated on its lateral surface and driven by a sinusoidal stress at one end, is solved using the Rayleigh-Ritz method. The displacement and temperature functions are expressed as series of polynomials. Successive approximations for the solution are compared with a solution obtained by a method of finite differences, and an estimate of the degree of accuracy as a function of the number of terms taken in the series is obtained. It is found that the approximate solution converges rapidly to the correct one.

A procedure to deal with various boundary conditions in the Rayleigh-Ritz method is discussed herein, namely, the method of artificial parameters. The method is then applied to solve a simple string problem; results obtained compare very nicely with exact solutions.

Application of Newton’s method to nonlinear vibration problems can lead to a sequence of nonhomogeneous ordinary differential equations with periodic coefficients. The form of the complementary solutions are known from Floquet theory. This paper suggests a method for avoiding “secular terms” that grow with time in the particular solution. The method consists of finding a single periodic solution of the complementary solutions and its adjoint. If the periodic solution exists, a frequency correction can be computed that eliminates secular terms. After the frequency correction, the rest of the particular solution is periodic and can be computed by the infinite determinant method or other numerical methods. In oversimplified terms, the procedure is to find the improved approximation to the period by variation of parameters and the next approximation to the amplitudes by undetermined coefficients which is a simpler computation than variation of parameters.

An algorithm is developed for the numerical solution of inhomogeneous linear twopoint boundary value problems using the method of invariant imbedding. The method handles the case in which the standard Riccati equation of the imbedding method fails to have a solution over the entire interval of interest. Very general sets of boundary conditions are imposed. While the development is for scalar problems only, all steps generalize readily to the case of matrix equations. An efficient program implementing the algorithm has been written and several examples are solved by the method. Some of these are of such a nature that more standard integration schemes provide unsatisfactory results.

A construction is given which allows the Hilbert space treatment of spline functions to be applied to the case of more than one variable, when the basic operator is a linear partial differential one. The particular case of the tensor product polynomial spline in two variables is then studied using a reproducing kernel, and its main properties, including the minimization ones, are deduced. A stable computational method is then given for this spline function, with certain point evaluation functionals. Finally, extensions are discussed, for more general linear functionals, for more general differential operators, and for more than two variables.

The general implicit finite-difference approximation of second order devised by von Neumann is applied to the initial boundary value problem for a somewhat generalized wave equation. By means of eigenvalue expansion it is shown that the method is uniform convergent of order $$\log \left( {\Delta x^{ - 1} } \right)O\left( {\Delta t^2 + \Delta x^2 } \right)$$ Δt, Δx mesh widths). Moreover, the convergence on the linet=T reveals to be proportional toT.

A comparative study is undertaken of 5- and 9-point discretizations of the linear source problem on a rectangular mesh, and of 3-point discretizations of its one-dimensional analogue. Traditional difference and (Richardson) extrapolation methods compare very favorably with varia- tional methods. Sample result: Courant's derivation of the standard 5-point formula for the Laplace equation from the Ritz variational method does not generalize to the Poisson or Helmholtz equation. 1. Introduction. We make below a comparative study of 5-point and 9-point discretizations of the differential equation of the linear source problem, (1) -V * (pVu) + qu = f, p(x, y) > O, q _ O, for piecewise smooth p, q and f on a rectangular mesh. We also study 3-point approximations to its one-dimensional analogue

The solution of Poisson's equation on the surface of a sphere may not exist unless the right side of the equation is perturbed. A method for perturbing is described which then admits a least squares solution. This least squares solution is obtained by the Fourier method which is economical in both computational time and storage.

A general method for the numerical analysis of frictionless nonconformable non-Hertzian contact of bodies of arbitrary shape is developed. Numerical difficulties arise because the solution is extremely sensitive to the manner in which one discretizes the governing integral equation. The difficulties were overcome by utilizing new techniques, referred to as the method of redundant field points (RFP) and the method of functional regularization (FR). The accuracy and efficiency of the methods developed were tested thoroughly against known solutions of Hertzian problems. To illustrate the power of the methods, a heretofore unsolved non-Hertzian problem (corresponding to the case of rounded indentors with local flat spots) has been solved.

The Cauchy problemu t =f(x, t, u, u x , u xx ),u(x, o)=?(x),x?R, is treated with the longitudinal method of lines. Existence, uniqueness, monotonicity and convergence properties of the line method approximations are investigated under the classical assumption that ? satisfies an inequality |?(x)| 0, and of Walter [11], who proved convergence in the case of nonlinear parabolic differential equations under the growth condition |?(x)

Two previously developed ray methods for conformal arrays on surfaces with variable curvature are further systematized and applied to arrays of equispaced identical slits on perfectly conducting parabolic cylindrical surfaces. The two methods differ in their definition of the canonical problems that are fundamental to the scattering process. In the first method, multiple scattering on the smooth cylinder surface accounts for interaction between isolated elements, while in the second, surface rays are defined on a local periodically loaded surface. The first method is valid under quite general conditions and yields reliable numerical results for mutual-coupling coefficients and radiation patterns. The second provides a simple explanation of a variety of coupling and radiation phenomena when the array possesses aspects of periodicity, and it is preferable for numerical calculation of mutual coupling in large arrays. These features are illustrated in detail on a particular example. The results obtained confirm the utility of the ray analysis in furnishing solutions for coupling coefficients and element patterns in conformal arrays, and they also demonstrate the fundamental insight gained by the local periodic-structure approach.

This paper describes a method of solving electromagnetic induction problems by means of the finite element technique. A variational equation associated with the differential equation for the vector potential is formulated and solved via the concept of finite elements. Sinusoidal driving currents and linear, isotropic, but inhomogeneous media are assumed. A typical example is included in order to illustrate the suggested solution technique.

The method of asymptotic integration, described in the paper, allows an iteration procedure for the successive determination of a approximate solution of three-demensional equations of geometrically nonlinear theory for the shell continuum. The classification of the shell problems is presented in terms of qualitative and quantitative characteristics of the shell data, and a possibility of estimating the influence of different types of external actions is indicated. On the basis of the results yielded by the asymptotic analysis, general information on the shell behavior may be deduced. A comparison of the predictions with the results of the numerical analysis is given for some plate and shell examples.

The simplest initial value problem for delay differential equations takes the form \[ {\text{(1)}}\qquad \dot y(t) = - \lambda y(t - \tau ),\quad t > 0;\qquad y(t) = \psi (t),\quad - \tau \leqq t \leqq 0.\] If the step size h is chosen so that $\tau = mh$ where m is an integer, then the linear multistep method $\{ \rho ,\sigma \} $ may be applied to (1) to yield an approximate solution $y^h $ defined at the gridpoints $t_j = jh$.It is known that $y(t) \to 0$ as $t \to \infty $ for all initial data $\psi $ iff $\lambda \in (0,{\pi / 2})$. In analogy with the concept of A-stability, the method $\{ \rho ,\sigma \} $ is called $DA_0 $-stable if $y^h (t_j ) \to 0$ as $j \to \infty $ for all initial data $\psi $ and all positive integers m. The properties of $DA_0 $-stable methods are studied.

The finite element method, using smooth splines as basis functions, applied to the model problem $u_t = cu_x $ with periodic data generates a differential-difference equation whose phase error is closely estimated and compared with the phase error of both explicit and high order implicit centered differencing. We also compute and compare the minimum work required to obtain a fixed error for several fully discrete schemes.

An a priori $L_\infty $ error estimate is established for continuous piecewise polynomial Galerkin approximations to solutions of linear two-point boundary value problems. This estimate is the best possible in that the order of convergence is optimal and the norm on the solutions cannot be weakened.

An alternate proof is presented of Kershaw’s result that the $L_\infty $-norm of the error in natural spline interpolation to a function $f \in C^4 [a,b]$ is $O(h^4 )$ in a closed subinterval which is asymptotic to $[a,b]$ as $h \to 0$. The case $f \in C^m [a,b],m = 2$ or 3, is also considered. These univariate results are then used to investigate natural bicubic spline interpolation over a rectangle $\mathcal{R}$. For $f \in C^m [\mathcal{R}]$, the $L_\infty $-norm of the error is shown to be $O(h^{m - 2} )$ throughout $\mathcal{R}$ and $O(h^m )$ in a closed subrectangle which is asymptotic to $\mathcal{R}$ as $h \to 0$. Arbitrary sequences of partitions are considered throughout.

An optical second degree iterative method for accelerating a basic linear stationary iterative method of first degree with zeal eigenvalues has been studied by Young [18], [19] and Young and Kincaid [20]. The derivation of this method is now established for the case when the eigenvalues are contained in an elliptical region in the complex plane. By extending results developed by Kublanowskaya (Faddeev–Faddeeva [1]) and N iethammer [8], this second-degree method is obtained through an application of conformal mapping and summation techniques.

It has been shown that the usual Galerkin procedure applied to first order hyperbolic equations does not yield the optimum $L^2 $ rate of convergence for all admissible finite-dimensional subspaces. Two methods are presented which overcome this difficulty.

A method is described for minimizing a continuously differentiable functionF(x) ofn variables subject to linear inequality constraints. It can be applied under the same general assumptions as any method of feasible directions. IfF(x) is twice continuously differentiable and the Hessian matrix ofF(x) has certain properties, then the algorithm generates a sequence of points which converges superlinearly to the unique minimizer ofF(x). No computation of secondorder derivatives is required.

Based on the least squares error criterion, a class of finite element is formulated for the numerical analysis of steady state viscous boundary layer flow problems. The method is essentially a discrete element-wise minimization of square and weighted residuals which arise from the attempts in approximately satisfying boundary layer equations. An iterative linearization scheme is developed to circumvent the mathematical difficulties posed by the non-linear boundary layer equations. It results in a process of successive least squares minimizations of residual errors arising from satisfying a set of linear differential equations. A mathematical justification for the method is presented. A major feature of the method lies in the linearization approach which renders non-linear differential equations amenable to linear least squares finite element analysis. Another important feature rests on the proposed finite element formulation which preserves the symmetric nature of finite element matrix equations through the use of the least squares error criterion. Numerical examples of viscous flow along a flat plate are presented to demonstrate the applicability of the method as well as to illuminate discussions on the theoretical aspects of the method.

This work describes a numerical method for the solution of the nonlinear Vlasov equation. The distribution function is expanded in a series of orthogonal polynomials, namely the Chebyshev polynomials. The conditions under which this expansion is valid are discussed. Recurrence effects are eliminated by formally adding a damping term to the eigenvalues of the truncated system. Nonlinear effects have been simulated by an amount of information corresponding to less than 500 “particles.”

1. The formulation of the problem of stability of vertical mining has been investigated here within the framework of the three-dimensional linearized theory of stability for small subcritical deformations, using the simplest model of rocks, i. e. a linear-elastic isotropic body. 2. Variational methods are developed for solving the problems and the solution is obtained for a mine of circular cross section with axisymmetric form of stability loss. 3. The asymptotic value of the critical load is computed for an arbitrary number of coordinate functions. 4. Using numerical methods and a computer sufficiently accurate value of critical loads is determined taking recourse to the three-dimensional linearized theory. The results are compared with those obtained by approximate methods.

A priori error estimates are established in the $L^2 $-norm for Galerkin approximations to the solution of a generalized wave equation. Optimal rates of convergence are established for several boundary conditions using both continuous and discrete Galerkin procedures.

It is shown that the rate of convergence of numerical approximations to the solution of hyperbolic partial differential equations is degraded when the solution being sought is not sufficiently smooth. A p th-order finite-difference scheme may give worse than p th-order convergence if the ( p + 1)st derivatives of the solution are not piecewise continuous. Spectral methods, which are normally expected to give infinite-order convergence, give only finite-order convergence if some derivative of the solution is not continuous. Near a discontinuity propagating along a characteristic of the differential equation, the truncation error of difference approximations is much larger on one side than on the other, and it oscillates in sign on the side where it is larger. (It may also oscillate on the other side of the discontinuity, depending on the order of the numerical scheme used.) Finally our analysis shows that, even if the true solution is not smooth, high-order schemes are more accurate than lower-order schemes.

A numerical-perturbation method is proposed for the determination of the nonlinear forced response of structural elements. Purely analytical techniques are capable of determining the response of structural elements having simple geometries and simple variations in thickness and properties, but they are not applicable to elements with complicated structure and boundaries. Numerical techniques are effective in determining the linear response of complicated structures, but they are not optimal for determining the nonlinear response of even simple elements when modal interactions take place due to the complicated nature of the response. Therefore, the optimum is a combined numerical and perturbation technique. The present technique is applied to beams with varying cross sections. ~ 4Y large-amplitude deflection of a beam or a plate which is restrained at its ends or along its edges results in some midplane stretching/One must account for this stretching with nonlinear strain-displacement relationships. The nonlinear equations of motion describing this situation were the basis of a number of earlier investigations and are the basis for the present paper as well. The purpose of the present paper is to present a new scheme for determining the response to a harmonic excitation. Emphasis is placed on the case when the frequency of the excitation is near a natural frequency. A convenient way to attack this nonlinear problem involves representing the deflection curve or surface with an expansion in terms of the linear, free-oscillation modes. The deflection is then determined in two steps. First, the damping, the forcing, and the nonlinear terms are deleted and the linear modes (eigenfunctions) and natural frequencies (eigenvalues) are determined. Second, the time-dependent coefficients in the expansion are obtained from a set of coupled, nonlinear, ordinary, second-order differential equations, the linear modes being used to determine the coefficients in these equations. (The procedure is described in detail in Sec. II.) Generally, one cannot obtain the linear modes analytically for structural elements having complicated boundaries and composition, and one cannot easily determine the character of the timedependent coefficients through numerical integration of the set of nonlinear equations. (The results obtained in the present numerical example are typical of the complicated manner in which the steady-state amplitudes of the various modes making up the response can vary with the amplitude and the frequency of the excitation.) Consequently, an optimal procedure involves a numerical method to determine the linear, free-oscillation modes and an analytical method to determine the time-dependent coefficients. The present procedure combines either a finiteelement or a finite-difference method with the method of multiple scales (see, for example, Ref. 1). The following brief review mentions representative examples of the work that was and is

We give a method for computing nontrivial solutions to the nonlinear partial differential equation ∇2Ψ + λ2sinh Ψ = 0, with Ψ = 0 on a square boundary. The method consists of a Newton-Raphson iteration, in which successive corrections to Ψ must satisfy a linearized partial differential equation. We give a direct solution algorithm for the linearized equation, which is suitable for small meshes. Using this method, we establish the nonuniqueness of solutions by finding six solutions for the same value of λ. Calculation of these solutions required from 4 to about 40 iterations each, depending upon the accuracy of the initial approximation. These solutions are the lowest-mode members of three classes of solutions possessing (1) rectangular, (2) quasicylindrical, and (3) diagonal symmetry.