Showing papers on "Discretization published in 1973"
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01 Jan 1973
TL;DR: The Discretization Methodology helps clarify the meaning of Consistency, Convergence, and Stability with Forward Step Methods and provides a guide to applications of Asymptotic Expansions in Even Powers of n.
Abstract: 1 General Discretization Methods.- 1.1. Basic Definitions.- 1.1.1 Discretization Methods.- 1.1.2 Consistency.- 1.1.3 Convergence.- 1.1.4 Stability.- 1.2 Results Concerning Stability.- 1.2.1 Existence of the Solution of the Discretization.- 1.2.2 The Basic Convergence Theorem.- 1.2.3 Linearization.- 1.2.4 Stability of Neighboring Discretizations.- 1.3 Asymptotic Expansions of the Discretization Errors.- 1.3.1 Asymptotic Expansion of the Local Discretization Error.- 1.3.2 Asymptotic Expansion of the Global Discretization Error.- 1.3.3 Asymptotic Expansions in Even Powers of n.- 1.3.4 The Principal Error Terms.- 1.4 Applications of Asymptotic Expansions.- 1.4.1 Richardson Extrapolation.- 1.4.2 Linear Extrapolation.- 1.4.3 Rational Extrapolation.- 1.4.4 Difference Correction.- 1.5 Error Analysis.- 1.5.1 Computing Error.- 1.5.2 Error Estimates.- 1.5.3 Strong Stability.- 1.5.4 Richardson-extrapolation and Error Estimation.- 1.5.5 Statistical Analysis of Round-off Errors.- 1.6 Practical Aspects.- 2 Forward Step Methods.- 2.1 Preliminaries.- 2.1.1 Initial Value Problems for Ordinary Differential Equations.- 2.1.2 Grids.- 2.1.3 Characterization of Forward Step Methods.- 2.1.4 Restricting the Interval.- 2.1.5 Notation.- 2.2 The Meaning of Consistency, Convergence, and Stability with Forward Step Methods.- 2.2.1 Our Choice of Norms in En and En0.- 2.2.2 Other Definitions of Consistency and Convergence.- 2.2.3 Other Definitions of Stability.- 2.2.4 Spijker'1 General Discretization Methods.- 1.1. Basic Definitions.- 1.1.1 Discretization Methods.- 1.1.2 Consistency.- 1.1.3 Convergence.- 1.1.4 Stability.- 1.2 Results Concerning Stability.- 1.2.1 Existence of the Solution of the Discretization.- 1.2.2 The Basic Convergence Theorem.- 1.2.3 Linearization.- 1.2.4 Stability of Neighboring Discretizations.- 1.3 Asymptotic Expansions of the Discretization Errors.- 1.3.1 Asymptotic Expansion of the Local Discretization Error.- 1.3.2 Asymptotic Expansion of the Global Discretization Error.- 1.3.3 Asymptotic Expansions in Even Powers of n.- 1.3.4 The Principal Error Terms.- 1.4 Applications of Asymptotic Expansions.- 1.4.1 Richardson Extrapolation.- 1.4.2 Linear Extrapolation.- 1.4.3 Rational Extrapolation.- 1.4.4 Difference Correction.- 1.5 Error Analysis.- 1.5.1 Computing Error.- 1.5.2 Error Estimates.- 1.5.3 Strong Stability.- 1.5.4 Richardson-extrapolation and Error Estimation.- 1.5.5 Statistical Analysis of Round-off Errors.- 1.6 Practical Aspects.- 2 Forward Step Methods.- 2.1 Preliminaries.- 2.1.1 Initial Value Problems for Ordinary Differential Equations.- 2.1.2 Grids.- 2.1.3 Characterization of Forward Step Methods.- 2.1.4 Restricting the Interval.- 2.1.5 Notation.- 2.2 The Meaning of Consistency, Convergence, and Stability with Forward Step Methods.- 2.2.1 Our Choice of Norms in En and En0.- 2.2.2 Other Definitions of Consistency and Convergence.- 2.2.3 Other Definitions of Stability.- 2.2.4 Spijker's Norm for En0.- 2.2.5 Stability of Neighboring Discretizations.- 2.3 Strong Stability of f.s.m..- 2.3.1 Perturbation of IVP 1.- 2.3.2 Discretizations of {IVP 1}T.- 2.3.3 Exponential Stability for Difference Equations on [0,?).- 2.3.4 Exponential Stability of Neighboring Discretizations.- 2.3.5 Strong Exponential Stability.- 2.3.6 Stability Regions.- 2.3.7 Stiff Systems of Differential Equations.- 3 Runge-Kutta Methods.- 3.1 RK-procedures.- 3.1.1 Characterization.- 3.1.2 Local Solution and Increment Function.- 3.1.3 Elementary Differentials.- 3.1.4 The Expansion of the Local Solution.- 3.1.5 The Exact Increment Function.- 3.2 The Group of RK-schemes.- 3.2.1 RK-schemes.- 3.2.2 Inverses of RK-schemes.- 3.2.3 Equivalent Generating Matrices.- 3.2.4 Explicit and Implicit RK-schemes.- 3.2.5 Symmetric RK-procedures.- 3.3 RK-methods and Their Orders.- 3.3.1 RK-methods.- 3.3.2 The Order of Consistency.- 3.3.3 Construction of High-order RK-procedures.- 3.3.4 Attainable Order of m-stage RK-procedures.- 3.3.5 Effective Order of RK-schemes.- 3.4 Analysis of the Discretization Error.- 3.4.1 The Principal Error Function.- 3.4.2 Asymptotic Expansion of the Discretization Error.- 3.4.3 The Principal Term of the Global Discretization Error.- 3.4.4 Estimation of the Local Discretization Error.- 3.5 Strong Stability of RK-methods.- 3.5.1 Strong Stability for Sufficiently Large n.- 3.5.2 Strong Stability for Arbitrary n.- 3.5.3 Stability Regions of RK-methods.- 3.5.4 Use of Stability Regions for General {IVP 1}T.- 3.5.5 Suggestion for a General Approach.- 4 Linear Multistep Methods.- 4.1 Linear k-step Schemes.- 4.1.1 Characterization.- 4.1.2 The Order of Linear k-step Schemes.- 4.1.3 Construction of Linear k-step Schemes of High Order.- 4.2 Uniform Linear k-step Methods.- 4.2.1 Characterization, Consistency.- 4.2.2 Auxiliary Results.- 4.2.3 Stability of Uniform Linear k-step Methods.- 4.2.4 Convergence.- 4.2.5 Highest Obtainable Orders of Convergence.- 4.3 Cyclic Linear k-step Methods.- 4.3.1 Stability of Cyclic Linear k-step Methods.- 4.3.2 The Auxiliary Method.- 4.3.3 Attainable Order of Cyclic Linear Multistep Methods.- 4.4 Asymptotic Expansions.- 4.4.1 The Local Discretization Error.- 4.4.2 Asymptotic Expansion of the Global Discretization Error, Preparations.- 4.4.3 The Case of No Extraneous Essential Zeros.- 4.4.4 The Case of Extraneous Essential Zeros.- 4.5 Further Analysis of the Discretization Error.- 4.5.1 Weak Stability.- 4.5.2 Smoothing.- 4.5.3 Symmetric Linear k-step Schemes.- 4.5.4 Asymptotic Expansions in Powers of h2.- 4.5.5 Estimation of the Discretization Error.- 4.6 Strong Stability of Linear Multistep Methods.- 4.6.1 Strong Stability for Sufficiently Large n.- 4.6.2 Stability Regions of Linear Multistep Methods.- 4.6.3 Strong Stability for Arbitrary n.- 5 Multistage Multistep Methods.- 5.1 General Analysis.- 5.1.1 A General Class of Multistage Multistep Procedures.- 5.1.2 Simple m-stage k-step Methods.- 5.1.3 Stability and Convergence of Simple m-stage k-step Methods.- 5.2 Predictor-corrector Methods.- 5.2.1 Characterization, Subclasses.- 5.2.2 Stability and Order of Predictor-corrector Methods.- 5.2.3 Analysis of the Discretization Error.- 5.2.4 Estimation of the Local Discretization Error.- 5.2.5 Estimation of the Global Discretization Error.- 5.3 Predictor-corrector Methods with Off-step Points.- 5.3.1 Characterization.- 5.3.2 Determination of the Coefficients and Attainable Order.- 5.3.3 Stability of High Order PC-methods with Off-step Points.- 5.4 Cyclic Forward Step Methods.- 5.4.1 Characterization.- 5.4.2 Stability and Error Propagation.- 5.4.3 Primitive m-cyclic k-step Methods.- 5.4.4 General Straight m-cyclic k-step Methods.- 5.5 Strong Stability.- 5.5.1 Characteristic Polynomial, Stability Regions.- 5.5.2 Stability Regions of PC-methods.- 5.5.3 Stability Regions of Cyclic Methods.- 6 Other Discretization Methods for IVP 1.- 6.1 Discretization Methods with Derivatives of f.- 6.1.1 Recursive Computation of Higher Derivatives of the Local Solution.- 6.1.2 Power Series Methods.- 6.1.3 The Perturbation Theory of Groebner-Knapp-Wanner.- 6.1.4 Groebner-Knapp-Wanner Methods.- 6.1.5 Runge-Kutta-Fehlberg Methods.- 6.1.6 Multistep Methods with Higher Derivatives.- 6.2 General Multi-value Methods.- 6.2.1 Nordsieck's Approach.- 6.2.2 Nordsieck Predictor-corrector Methods.- 6.2.3 Equivalence of Generalized Nordsieck Methods.- 6.2.4 Appraisal of Nordsieck Methods.- 6.3 Extrapolation Methods.- 6.3.1 The Structure of an Extrapolation Method.- 6.3.2 Gragg's Method.- 6.3.3 Strong Stability of MG.- 6.3.4 The Gragg-Bulirsch-Stoer Extrapolation Method.- 6.3.5 Extrapolation Methods for Stiff Systems.
501 citations
TL;DR: In this paper, the Newmark family of second-order difference approximations is compared with the original or extended Wilson and Houboult methods for the direct time integration of the spatially discretized equations of linear elastodynamics.
260 citations
TL;DR: In this article, a method for solving numerically the two-dimensional (2D) semiconductor steady-state transport equations is described, where Poisson's equation and the two continuity equations are discretized on two networks of different rectangular meshes.
Abstract: A method for solving numerically the two-dimensional (2D) semiconductor steady-state transport equations is described. The principles of this method have been published earlier [1]. This paper discusses in detail the method and a number of considerable improvements. Poisson's equation and the two continuity equations are discretized on two networks of different rectangular meshes. The 2D continuity equations are approximated by a set of difference equations assuming that the hole and electron current density components along the meshlines are constant between two neighboring meshpoints in a way similar to that used by Gummel and Scharfetter [2] for the one-dimensional (1D) continuity equations. The resulting difference approximations have generally a much larger validity range than the conventional difference formulations where it is assumed that the change in electrostatic potential between two neighboring points is small compared with k T/q . Therefore, a much smaller number of meshpoints is necessary than for the conventional difference approximations. This reduces considerably the computation time and the required memory space. It will be shown that the matrix of the coefficients of this set of difference equations is always positive definite. This is an important property and guarantees convergence and stability of the numerical solution of the continuity equations. The way in which the difference approximations for the continuity equations are derived gives directly consistent expressions for the current densities that can be used for calculating the currents. In order to demonstrate the kind of solutions obtainable, steady-state results for a bipolar n-p-n silicon transistor are presented and discussed.
196 citations
TL;DR: The boundary integral equation (BE) method as mentioned in this paper is based on a mathematical formulation which reduces the dimensionality of a problem by relating surface tractions to surface displacements and finds the stresses at any point are then found by direct quadrature from the entirety of surface data.
144 citations
TL;DR: In this paper, an implicit finite difference method for the multidimensional Stefan problem is discussed, where the classical problem with discontinuous enthalpy is replaced by an approximate Stefan problem with continuous piecewise linear enthpy.
Abstract: An implicit finite difference method for the multidimensional Stefan problem is discussed. The classical problem with discontinuous enthalpy is replaced by an approximate Stefan problem with continuous piecewise linear enthalpy. An implicit time approximation reduces this formulation to a sequence of monotone elliptic problems which are solved by finite difference techniques. It is shown that the resulting nonlinear algebraic equations are solvable with a Gauss-Seidel method and that the discretized solution converges to the unique weak solution of the Stefan problem as the time and space mesh size approaches zero.
136 citations
TL;DR: In this article, upper and lower bounds on the spectral and maximum norms of stiffness, flexibility and mass matrices generated from regular and irregular meshes of finite elements were established for second and fourth order problems in one, two and three dimensions discretized with linear, triangular and tetrahedronal elements.
52 citations
TL;DR: Two methods of simulating continuous-data systems by sampled-data models are presented; one relies on the optimization of the sampled- data system with a quadratic performance index, and another relies on a point-by-point comparison method.
41 citations
TL;DR: A numerical example based on an eight-storm record is included, where the objective function for the optimization procedure is the sum of squared deviations between observed and predicted output ordinates for all storms included in the record.
Abstract: Adopting a discretization scheme for input, output, and kernel functions of second-order Volterra series representation of the surface runoff system leads to a set of linear equations for the unknown ordinates of the two-kernel functions. The equations are solved by an iterative descent optimization procedure taking into account also additional constraining equations derived from the properties of the kernel functions and the definition of the system adopted. The objective function for the optimization procedure is the sum of squared deviations between observed and predicted output ordinates for all storms included in the record. The inclusion of the constraining equations is done by an iterative scheme based on the penalty function method. The algorithm adopted for the solution converges to the optimal solution in a finite number of iterations, and its rate of convergence is independent of initial values adopted for the unknowns. A numerical example based on an eight-storm record is included.
37 citations
TL;DR: Discretizing certain discrete-time, uncountable-state dynamic programs such that the respective solutions to a sequence of discretized versions converge uniformly to the solution of the original problem is shown.
Abstract: Discretizing certain discrete-time, uncountable-state dynamic programs such that the respective solutions to a sequence of discretized versions converge uniformly to the solution of the original problem is shown. Via a random perturbation device, we apply our general approach to a separable nonconvex program.
33 citations
TL;DR: Using the equivalent-quadrature relation between diagonalization of the s-wave kinetic energy H/sup o/ in a Laguerre-type basis and Chebyschev quadrature of the second kind, a procedure is given by which potential-scattering phase shifts may be constructed from approximations to the Fredholm determinant constructed using only square-integrable (L/sup 2/) functions as mentioned in this paper.
Abstract: Using the equivalent-quadrature'' relation between diagonalization of the s-wave kinetic energy H/sup o/ in a Laguerre-type basis and Chebyschev quadrature of the second kind, a procedure is given by which potential-scattering phase shifts may be constructed from approximations to the Fredholm determinant constructed using only square-integrable (L/sup 2/) functions. For the problem of electron-hydrogen scattering in the static approximation, highly accurate phase shifts are obtained over a continuous range of energies from a single major computational step. (auth)
TL;DR: The carrying over of normality from the element to the system is proved by means of virtual work equations, and alternatively, by deriving a suitable matrix description of the system incremental laws that leads to sufficient conditions for stability and for existence and uniqueness of the incremental response.
Abstract: Essential features include normality or lack of normality of the inelastic strain or displacement rates to a meaningful surface, convexity or concavity of this surface, stability, and uniqueness of the incremental response. The correlation between system features and those of its constituent elements is studied in the presence of significant geometric effects on equilibrium. The system is regarded as an assembly of a discrete number of constituents via finite element discretization; consequently, algebraic description is adopted. The carrying over of normality from the element to the system is proved by means of virtual work equations, and alternatively, by deriving a suitable matrix description of the system incremental laws. This description combined with some recent results in mathematical programming, leads to sufficient conditions for stability and for existence and uniqueness of the incremental response. These conditions are formulated for both individual elements and systems and can be used for actual calculations. Convexity does not in general carry over from element to system in the presence of significant geometry changes
TL;DR: In this article, a mixed finite-difference scheme is presented for the stress and free vibration analysis of simply supported nonhomogeneous and layered orthotropic thick plates, which is based on the linear, three-dimensional theory of orthotropic elasticity and a Fourier approach is used to reduce the governing equations to six firstorder ordinary differential equations in the thickness coordinate.
TL;DR: The method of analysis is primarily concerned with the determination of the forces and displacements that result from live loads and temperature changes; however, it can also be used to determine the initial dead load configuration.
TL;DR: In this article, an efficient digital computer procedure and a related numerical algorithm are presented for the solution of quadratic matrix equations associated with free vibration analysis of structures, which enables accurate and economical analysis of natural frequencies and associated modes of discretized structures.
Abstract: An efficient digital computer procedure and the related numerical algorithm are presented herein for the solution of quadratic matrix equations associated with free vibration analysis of structures. Such a procedure enables accurate and economical analysis of natural frequencies and associated modes of discretized structures. The numerically stable algorithm is based on the Sturm sequence method, which fully exploits the banded form of associated stiffness and mass matrices. The related computer program written in FORTRAN V for the JPL UNIVAC 1108 computer proves to be substantially more accurate and economical than other existing procedures of such analysis. Numerical examples are presented for two structures - a cantilever beam and a semicircular arch.
01 Sep 1973
TL;DR: In this article, finite element incremental formulations for nonlinear static and dynamic analysis are reviewed and derived starting from continuum mechanics principles, including large displacements, large strains, material nonlinearities and nonconservative forces.
Abstract: Starting from continuum mechanics principles, finite element incremental formulations for nonlinear static and dynamic analysis are reviewed and derived. No new formulation is presented. The aim in the report is a consistent summary, comparison, and evaluation of the two formulations, which are used in the general purpose nonlinear static and dynamic analysis program NONSAP. The general formulations include large displacements, large strains, material nonlinearities and nonconservative forces. For specific solutions in the report, elastic and hyperelastic materials only are considered. The numerical solution of the continuum mechanics equations is achieved in NONSAP using isoparametric finite element discretization. The specific matrices which need to be calculated in the formulations are discussed. (Author)
TL;DR: In this paper, a numerical technique is presented which permits a computer solution of the complete set of time dependent partial differential equations governing bipolar semiconductor behavior, which does not require any of the often made assumptions and approximations such as abrupt junctions, quasi-neutrality or restricted injection levels.
Abstract: A numerical technique is presented which permits a computer solution of the complete set of time dependent partial differential equations governing bipolar semiconductor behavior. The scheme does not require any of the often made assumptions and approximations such as abrupt junctions, quasi-neutrality or restricted injection levels. The resulting solution describes device terminal properties and gives a detailed account of internal parameters as a function of time and distance. The present method is distinguished from others by the use of the technique of quasilinearization which converts the nonlinear boundary value problem into a linear form. The latter is solved iteratively, yielding highly stable and convergent solutions. The method uses the complete form of Poisson's equation and an implicit time advancement formulation to generate a stable time trajectory of the solution. The Scharfetter-Gummel spatial discretization scheme is used for greater accuracy. Computer and experimental results are given and compared for the transient behavior of a p+nn+ diode structure. Both forward and reverse transients are described.
TL;DR: In this article, a modified multilocal difference scheme is presented for free vibration analysis of nonhomogeneous orthotropic noncircular cylindrical shells with simply supported curved edges.
Abstract: A modified multilocal difference scheme is presented for the free vibration analysis of nonhomogeneous orthotropic noncircular cylindrical shells with simply supported curved edges. The problem is formulated in terms of 10 first-order ordinary differential equations and in the finite-difference discretization two interlacing grids are used for the different fundamental unknowns in such a way as to reduce both the local discretization error and the bandwidth of the resulting finite-difference field equations. Numerical studies are presented for the effects of reducing the interior and boundary discretization errors and of mesh refinement on the accuracy and convergence of solutions. It is shown that the proposed scheme, in addition to a number of other advantages, leads to more accurate results than those obtained by other multilocal and ordinary difference schemes reported here-to-fore in the literature.
TL;DR: In this paper, the nonconformal contact problem in three-dimensional elastostatics has been formulated and successfully solved using the functional regularization method of Tychonov.
TL;DR: In this paper, the authors describe a numerical technique designed to overcome the difficulties encountered in the approximate solution of two-dimensional discretized field problems when there are singularities arising from sharp corners, discontinuities, point sources, etc.
Abstract: This paper contains a description of a numerical technique designed to overcome the difficulties encountered in the approximate solution of two-dimensional discretized field problems when there are singularities arising from sharp corners, discontinuities, point sources, etc. The essence of the method is to modify the discretization by incorporating in the numerical computations the known analytical form of the singularities. The method is applicable to a wide variety of linear two-dimensional field problems, and is illustrated herein with plane elastic problems.
01 Sep 1973
TL;DR: In this article, a method of analyzing nonlinear static and dynamic responses of deformable solids has been developed based on an incremental variational formulation using the Lagrangian mode of description.
Abstract: : A method of analyzing nonlinear static and dynamic responses of deformable solids has been developed based on an incremental variational formulation using the Lagrangian mode of description. The material nonlinearity due to plasticity or viscoplasticity as well as the geometric nonlinearity due to large displacements are considered. The equations of motion are obtained in a linearized incremental form using the principle of virtual work and solved using step-by-step numerical integration procedures. Equilibrium check is made at the end of each step and the residual forces are added to the next increment for improved accuracy over the pure incremental method. For elastic-plastic solutions the flow theory of plasticity is used along with the von Mises yield condition for isotropically hardening materials. The viscoplastic constitutive theory is also in the form of an associated flow law and capable of considering strain rate sensitive behavior. The viscoplastic strains are taken into account using an initial strain formulation. The discretization of the structure is achieved by the use of degenerate isoparametric finite elements and the computer codes that have been developed are capable of analyzing large axisymmetric deformations of shells of revolution. (Modified author abstract)
TL;DR: In this paper, the authors considered the problem of generalized rational approximation on a compact metric space, where the denominators were assumed to be uniformly bounded from below by a positive constant, and they showed that, if we discretize the problem, i.e. replace T by a sequence of finite subsetsT m of T converging to T, then the corresponding discrete minimal distances converge to the minimal distance of the given problem and there is a subsequence of discrete best approximants which converges uniformly to the solution of the initial problem.
Abstract: In this paper we consider the problem of generalized rational approximation on a compact metric spaceT where we assume the denominators to be uniformly bounded from below by a positive constant. We show that, if we discretize the problem, i. e. replaceT by a sequence {T m } of finite subsetsT m ofT converging toT, then the corresponding discrete minimal distances converge to the minimal distance of the given problem and there is a subsequence of discrete best approximants which converges uniformly to the solution of the initial problem. If this is uniquely solvable, then each sequence of discrete best approximants converges uniformly to the solution of the initial problem.
07 May 1973
TL;DR: The resolution of optimal control problems for systems described by partial differential equations leads, after complete (or semi) discretization, to large-scale optimization problems on models described by difference (or differential) equations that are often not easy to solve directly.
Abstract: The resolution of optimal control problems for systems described by partial differential equations leads, after complete (or semi) discretization, to large-scale optimization problems on models described by difference (or differential) equations that are often not easy to solve directly.
01 Sep 1973
TL;DR: In this article, a user-oriented subroutine package is built around a highly stable explicit integration algorithm for solution of large order systems of ordinary differential equations, as result from discretization of initial-boundary value problems in continuum mechanics.
Abstract: A user-oriented subroutine package is built around a highly stable explicit integration algorithm for solution of large order systems of ordinary differential equations, as result from discretization of initial-boundary value problems in continuum mechanics. Fast and accurate solutions, for problems in laminar and turbulent, two and three-dimensional viscous flow fields and multi-dimensional transient heat transfer, are presented using this algorithm, as embodied within a general purpose finite element computer program.
01 Jan 1973
TL;DR: Structural optimization, which is the goal of such studies, receives nowadays a good deal of attention; its costeffective implementation will largely depend on the speed and accuracy with which the more classical problems can be solved, namely the determination of its statical and dynamical response to external loads or displacements of supports.
Abstract: Elasticity theory, even in its simplest guise, linear with respect to geometrical deformations and with respect to material properties, has few so-called “closed form” solutions. The need for solving numerically elasticity problems and the advent of fast and powerful computing machines seem even to have partly robbed such types of solutions of their interest. It is now often easier and faster to produce numerical results of a given problem by using modern discretization methods than to have to compute the special functions involved in a closed form presentation. In the end the value of such presentations lies in the ease with which they can be manipulated to exhibit the influence of the design parameters, whereas a similar computer study of the influence of design is still a long and costly process. Structural optimization, which is the goal of such studies, receives nowadays a good deal of attention; its costeffective implementation will largely depend on the speed and accuracy with which the more classical problems can be solved, namely the determination, for a given design, of its statical and dynamical response to external loads or displacements of supports.
TL;DR: In this article, a method of analysis using finite element techniques is presented for second order, mixed boundary value problems in the plane, focusing computational effort on specific points in the domain and providing absolute solution error bounds at those points by applying the hypercircle method.
TL;DR: In this paper, the authors unify a number of continuity, perturbation, and discretization properties of best approximations in normed linear spaces or in metric or semimetric spaces.
Abstract: The purpose of this paper is to unify a number of continuity, perturbation, and discretization properties of best approximations in normed linear spaces or in metric or semimetric spaces. It extends work of Aubin and, more specifically, of Daniel.
TL;DR: In this paper, a new method is introduced for solving free-boundary problems for the heat equation, which is shown to be uniformly convergent both for first type and nonlinear second type boundary conditions.
Abstract: A new method is introduced for solving free-boundary problems for the heat equation. The method is shown to be uniformly convergent both for first type and nonlinear second type boundary conditions. Approximate solutions are obtained by discretization of the time variable and recursive solution at each time step of classical heat conduction linear problems in slabs of known thickness. An error estimate is given. Numerical tests are presented in a few cases in which exact solutions are known.
01 Jan 1973
TL;DR: In this paper, generalized matrix inverse techniques for local approximations of operator equations are presented, which are based on the Galerkin approach and the Rayleigh-Ritz analysis.
Abstract: Publisher Summary This chapter presents the generalized matrix inverse techniques for local approximations of operator equations. Considerable advance has been made in the practical implementation of the Galerkin and the Rayleigh–Ritz methods to equations of importance to engineering and physics. The prime features of this advance have been the use of localized basis functions and the resulting discretization processes that are amenable to computer instruction. In terms of the Galerkin approach, the discretization process has resulted in approximate general solutions of differential equations, while with Rayleigh–Ritz analysis it has fostered the conventional finite element method. As with any discretization process, both of these methods consist of two separate steps. First, an approximation of the functions comprising the domain space of the operator must be established and, second, a numerical equivalent of the operator acting on this space must be devised. Unfortunately, all existing developments of these methods emphasize the second of these steps. Accordingly, while methods furnishing good solutions of differential equations are available, uncertainties and difficulties in the development and the use of legitimate approximation functions for the domain space of an operator are often encountered.
TL;DR: In this article, the Halyo-McAlpine discrete model for product quantization is reconstructed using an error variable set having an even number of members, and the modified error set is shown to be more consistent with binary data formats such as two's complement.
Abstract: The Halyo-McAlpine discrete model for product quantization is reconstructed using an error variable set having an even number of members. It is shown in this case that the noise variance associated with a continuous model becomes an upper rather than a lower bound on the variance of the product quantization noise. The modified error set is shown to be more consistent with binary data formats such as two's complement.