Linear Elliptic Partial Differential Equations Research Articles

Multiresolution green's function methods for interactive simulation of large-scale elastostatic objects

We present a framework for low-latency interactive simulation of linear elastostatic models, and other systems arising from linear elliptic partial differential equations, which makes it feasible to interactively simulate large-scale physical models. The deformation of the models is described using precomputed Green's functions (GFs), and runtime boundary value problems (BVPs) are solved using existing Capacitance Matrix Algorithms (CMAs). Multiresolution techniques are introduced to control the amount of information input and output from the solver thus making it practical to simulate and store very large models. A key component is the efficient compressed representation of the precomputed GFs using second-generation wavelets on surfaces. This aids in reducing the large memory requirement of storing the dense GF matrix, and the fast inverse wavelet transform allows for fast summation methods to be used at run-time for response synthesis. Resulting GF compression factors are directly related to interactive simulation speedup, and examples are provided with hundredfold improvements at modest error levels. We also introduce a multiresolution constraint satisfaction technique formulated as a hierarchical CMA, so named because of its use of hierarchical GFs describing the response due to hierarchical basis constraints. This direct solution approach is suitable for hard real-time simulation since it provides a mechanism for gracefully degrading to coarser resolution constraint approximations. The GFs' multiresolution displacement fields also allow for run-time adaptive multiresolution rendering.

Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients

We present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients. Based on the Maple software package, we approximate the leading terms in the truncation error of the Taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation. A test problem is solved numerically to validate the derived fourth order compact difference scheme. This symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations.

REGULARITY RESULTS FOR LINEAR ELLIPTIC PROBLEMS RELATED TO THE PRIMITIVE EQUATIONS

The authors study the regularity of solutions of the GFD-Stokes problem and of some second order linear elliptic partial differential equations related to the Primitive Equations of the ocean. The present work generalizes the regularity results in [18] by taking into consideration the non-homogeneous boundary conditions and the dependence of solutions on the thickness e of the domain occupied by the ocean and its varying bottom topography. These regularity results are important tools in the study of the PEs (see e.g. [6]), and they seem also to possess their own interest.

Non-coercive Linear Elliptic Problems

We study here some linear elliptic partial differential equations (with Dirichlet, Fourier or mixed boundary conditions), to which convection terms (first order perturbations) are added that entail the loss of the classical coercivity property. We prove the existence, uniqueness and regularity results for the solutions to these problems.

The efficiency of the p‐version finite element method in a distributed computing environment

AbstractThe p‐version of the finite element method is now commonly considered to be a very accurate discretization method for linear elliptic partial differential equations, but many researchers still doubt the efficiency of this method, when compared to the classical h‐version and applied to more complex problems. This paper will first discuss some general considerations about the efficiency of a numerical method and then present results on an evaluation of the p‐version. It will be shown, that there are many special techniques being applicable to the p‐version, yielding a well‐performing and robust method. This will be demonstrated on several examples, including non‐linear problems and a parallel implementation on a workstation cluster. A client–server software structure for an efficient integration of CAD and FEA using a strict separation of geometric and non‐geometric aspects will also be outlined. Copyright © 2001 John Wiley & Sons, Ltd.

Me´thode des E´le´ments Finis en Me´canique des Structures

7R3. Me´thode des E´le´ments Finis en Me´canique des Structures. Finite Element Methods in the Mechanics of Structures. (French). - T Gmur. Presses Polytech et Univ Romandes, Lausanne. 2000. 252 pp. ISBN 2-88074-461-X. Reviewed by VD Radulescu (Dept of Math, Univ of Craiova, 13, St AI Cuza, Craiova, 1100, Romania).This textbook deals with the study of some elementary basic problems raised by Applied Mechanics. The book is intended for a large audience: students in Applied Sciences, researchers, engineers, etc. Let Ω be a smooth bounded domain in RN. The problems studied in the present work are of the following (weak) form: find a vector field u such that ∫Ω vT[Du-f]dΩ=0 ∀v,where D is a certain linear differential operator and f is a given vector field. Throughout the work, the study is limited to the case of linear elliptic partial differential equations of second order. In many cases, the exact solution of this problem (if it exists!) cannot be found explicitly. That is why several methods have been developed in Numerical Analysis in order to approximate the solutions of wide classes of differential equations. The Finite Element Method was introduced in the 50s, and it can be viewed as an extension of the classical Galerkin Method. The Finite Element Method consists in the decomposition of Ω into a finite number of subdomains, called finite elements. The solution of the initial problem is then found as an assemblage of functions with compact support. After a short introduction, the author introduces in Chapter 2 the strong and the weak mathematical formulation in the one-dimensional case. This elementary frame-work enables the author to explain carefully how numerical methods can be applied for approximating the solution. The Finite Element Method is based in this simple case by an integral approach. In the next chapter, the weak formulation of the approached problem is developed, and one-dimensional linear finite problems of second degree are considered. Chapter 4 is devoted to the presentation of the weak formulation to linear bidimensional problems of the second degree. The study is developed on the model of the heat-transfer equation. In this case, the approach is based on planar finite elements. The author develops the mathematical treatment of the problem, which includes convergence and numerical integration. A general situation is treated in Chapter 5, which includes applications in Linear Elasticity for two or three dimensions. The last chapter is devoted to some applications of the Finite Element Method. Here, the numerical solutions are confronted to experimental results. In this chapter, but also throughout the book, the exposition simplifies many results that have so far only been accessible in several journal articles. In short, there is an enormous wealth of content provided by the author, much of it cannot be found in any single source. Certainly, many techniques related to the Finite Element Method are discussed, so the title is unquestionably appropriate. The book and its extensive bibliography (100 titles strictly related to the subject) should serve as a tremendous reference for all researchers in the field and certainly belongs on the bookshelf next to other references on the Finite Element Method. One of the reasons why the author is able to cover such vast material in the book is that he tends to develop the main ideas exactly to the right degree of generality that he needs. Moreover, the treatment is always done by progressing from the special to the general case, essentially avoiding pedantic verifications. Sometimes the proof is even given through the right picture. In this case, the author has really saved the reader from a lot of unnecessary heavy notation. Moreover, the notation is not only simple, but also everywhere consistent. One basic question this reviewer would like to answer is whether this book is meant for students. My feeling is that, first of all, it is very nice for students to see so many concrete examples and pictures. The reader here is well motivated by simple, but illuminating examples before being faced with general notions. This is due quite systematically in the book, where examples and pictures always precede the proofs and allow those to be easily stated and easily understood. This reviewer concludes that Me´thode des E´le´ments Finis en Me´canique des Structures is an attractive book, full of concrete information, which gives a clear and lucid view of the current knowledge on the resolution of differential equations with the Finite Element Method.

Modified multigrid for 3D elliptic equations with cross-derivatives

A portable Fortran code that solves the general nonseparable three-dimensional linear elliptic partial differential equation (PDE) with cross-derivative terms on a rectangular region is described. Boundary conditions can be any combination of periodic, specified, or mixed derivative. A multigrid scheme, modified to handle complexities introduced by cross-derivative terms and nonnormal derivative components in boundary conditions, is utilized. Successful application is illustrated with an analytically prescribed abstract example and simulation of an idealized geophysical fluid flow.

Saint-Venant End Effects in Antiplane Shear for Functionally Graded Linearly Elastic Materials

The purpose of this research is to investigate the influence of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. The work is motivated by the recent research activity on functionally graded materials (FGMs) i.e., materials with continuously varying properties tailored to satisfy specific engineering applications. In the framework of antiplane shear deformations of a semi-infinite strip, the mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to a Dirichlet boundary value problem for a second-order linear elliptic partial differential equation with variable coefficients. In previous work, the variable coefficient (the shear modulus) was assumed to be strictly positive on the closed semi-infinite strip. The authors relax this assumption to allow for the shear modulus to be zero on either long side of the strip. Thus, the results are applicable to FGMs that are graded continuously from zero shear modulus at one side of the strip. Lower bounds for the rate of decay of end effects are obtained, which allow for assessing the influence of material inhomogeneity. The results are illustrated using several specific material models.

Beltrami operators in the plane

We determine optimal Lp-properties for the solutions of the general nonlinear elliptic system in the plane of the form f$\overline{z}$=H(z, fz), h∈ Lp(C), where H is a measurable function satisfying |H(z,w1)−H(z,w2)|≤ k|w1−w2| and k is a constant k<1. We also establish the precise invertibility and spectral properties in Lp(C) for the operators I−Tμ, I−μT, and T−μ, where T is the Beurling transform. These operators are basic in the theory of quasi-conformal mappings and in linear and nonlinear elliptic partial differential equations (PDEs) in two dimensions. In particular, we prove invertibility in Lp(C) whenever 1+||μ||∞ <p<1+1/||μ||∞. We also prove related results with applications to the regularity of weakly quasiconformal mappings.

Recovery of the boundary data for a linear second order elliptic problem with a nonlocal boundary condition

We study a linear elliptic partial differential equation of second order in a bounded domain ? ? R N , with nonstandard boundary conditions on a part ? of the boundary ??. Here, neither the solution nor its normal derivative are prescribed pointwise. Instead, the average of the solution over ? is given and the normal derivative along ? has to follow a prescribed shape function, apart from an additive (unknown) constant. We prove the well-posedness of the problem and provide a method for the recovery of the unknown boundary data.

Stabilized continuation methods for boundary value problems

Inspired by the classical control theory for linear dynamical systems, we present an analytical continuation algorithm with inherent error stabilization. Numerical error is stabilized quite the same way as controlling the tracking error dynamics in system theory. Simulations demonstrate the efficacy of the proposed method wherein the stabilized continuation method always converges to the unique solution (if it exists) or any feasible solution (in the case of non-unique solutions). Further, we find new applications for this method in solving second order linear and non-linear elliptic partial differential equations with linear and non-linear boundary conditions. Illustrative numerical examples deal with stiff non-linear systems arising out of various physical applications.

Computing singular solutions of elliptic boundary value problems in polyhedral domains using the p-FEM

Numerical methods for computing singular solutions of linear second order elliptic partial differential equations (Laplace and Elasticity problems) in polyhedral domains are presented. The singularities may be caused by edges, vertices, or abrupt changes in material properties or boundary conditions. In the vicinity of the singular lines or points the solution can be represented by an asymptotic series, composed of eigen-pairs and their amplitudes. These are of great interest from the point of view of failure initiation because failure theories directly or indirectly involve them. This paper addresses a general method based on the modified Steklov formulation for computing the eigen-pairs and a dual weak formulation for extracting the amplitudes numerically using the p-version of the finite element method. The methods are post-solution operations on the finite element solution vector and have been shown in a two dimensional setting to be super-convergent.

On the method of functional equations and the performance of desingularized boundary element methods

A correspondence is made between the reciprocal relation for linear elliptic partial differential equations and the Riesz integral representation. The former relates the boundary distributions and appropriate normal fluxes of two arbitrary solutions, and the latter expresses a continuous linear functional in terms of an integral involving a representing function. When sufficient regularity conditions are met, the representing function is identified with the unknown boundary distribution. In principle, the representing function may be expressed in terms of the images of a complete set of orthonormal basis functions with known normal fluxes, as suggested by Kupradze [Kupradze VD. On the approximate solution of problems in mathematical physics. Russ Math Surv 1967; 22: 59–107]; in practice, the representing function is computed by solving integral equations using boundary element methods. The basic procedure involves expressing the representing function in terms of finite-element or other basis functions, and requiring the satisfaction of the reciprocal relationship with a suitable set of test functions such as Green's functions and their dipoles. When the singular points are placed at the boundary, we obtain the standard boundary integral equation method. When the singular points are placed outside the domain of solution, we obtain a system of functional equations and associated class of desingularized boundary integral methods. When sufficient regularity conditions are met and the test functions comprise a complete set, then in the limit of infinite discretization the numerical solution converges to the unknown boundary distribution. An overview of formulations is presented with reference to Laplace's equation in two dimensions. Numerical experimentation shows that, in general, the solution obtained by desingularized methods becomes increasingly less accurate as the singular points of Green's functions move farther away from the boundary, but the loss of accuracy is significant only when the exact solution shows pronounced variations. Exceptions occur when the integral equation does not have a unique solution. In contrast, and in agreement with previous findings, the condition number of the linear system increases rapidly with the distance of the singular points from the boundary, to the extent that a dependable solution cannot be obtained when the singularities are located even a moderate distance away from the boundary. The desingularized formulation based on Green's function dipoles is superior in accuracy and reliability to the one that uses Green functions. The implementation of the method to the equations of elastostatics and Stokes flow are also discussed.

A posteriori estimators for nonlinear elliptic partial differential equations

Many works have reported results concerning the mathematical analysis of the performance of a posteriori error estimators for the approximation error of finite element discrete solutions to linear elliptic partial differential equations. For each estimator there is a set of restrictions defined in such a way that the analysis of its performance is made possible. Usually, the available estimators may be classified into two types, i.e., the implicit estimators (based on the solution of a local problem) and the explicit estimators (based on some suitable norm of the residual in a dual space). Regarding the performance, an estimator is called asymptotically exact if it is a higher-order perturbation of a norm of the exact error. Nowadays, one may say that there is a larger understanding about the behavior of estimators for linear problems than for nonlinear problems. The situation is even worse when the nonlinearities involve the highest derivatives occurring in the PDE being considered (strongly nonlinear PDEs). In this work we establish conditions under which those estimators, originally developed for linear problems, may be used for strongly nonlinear problems, and how that could be done. We also show that, under some suitable hypothesis, the estimators will be asymptotically exact, whenever they are asymptotically exact for linear problems. Those results allow anyone to use the knowledge about estimators developed for linear problems in order to build new reliable and robust estimators for nonlinear problems.

Asymptotic a posteriori finite element bounds for the outputs of noncoercive problems: the Helmholtz and Burgers equations

We describe an a posteriori finite element procedure for the efficient computation of lower and upper estimators for linear-functional outputs of noncoercive linear and semilinear elliptic second-order partial differential equations. Under a relatively weak hypothesis related to the relative magnitude of the L 2 and H 1 errors of the reconstructed solution, these lower and upper estimators converge to the true output from below and above, respectively, and thus constitute asymptotic bounds. In numerical experiments we find that our hypothesis is satisfied once the finite element triangulation even roughly resolves the structure of the exact solution, and thus, in practice, the bounds prove quite reliable. Numerical results are presented for the one-dimensional Helmholtz equation and for the Burgers equation.

Fundamental solutions for second order linear elliptic partial differential equations

This paper is concerned with outlining some fundamental solutions and Green's functions for a system of second order linear elliptic partial differential equations in two independent variables. The fundamental solution and a number of Green's functions are given in relatively elementary closed form for some cases when the coefficients in the equations are constant. When the coefficients are variable the fundamental solution is obtained for some particular classes of equations.

Optimal control of fluid flow in soil I. Deterministic case

We study the numerical aspect of the optimal control of problems governed by a linear elliptic partial differential equation (PDE). We consider here the gas flow in porous media. The observed variable is the flow field we want to maximize in a given part of the domain or its boundary. The control variable is the pressure at one part of the boundary or the discharges of some wells located in the interior of the domain. The objective functional is a balance between the norm of the flux in the observation region and the costs due to the control variables. We consider several geometric configurations of the control and the observation variables, and we make use of different objective functionals. We take advantage of the linearity of the flux w.r.t. the control variable to significantly reduce the computational effort and to deduce the optimal controls of wide class of objective functionals. In this paper we consider the deterministic case where the model parameters are given in the whole domain.

ADI Methods for Cubic Spline Collocation Discretizations of Elliptic PDE

This paper presents the formulation, analysis, and implementation of alternating direction implicit (ADI) methods for solving the linear system of algebraic equations that arise from the discretization of multidimensional linear elliptic partial differential equations (PDEs). The theoretical analysis is carried out for a broad class of PDE problems. Numerical experiments confirm the theoretically determined characteristics of the ADI iterative schemes. The computational effectiveness of the proposed schemes is shown through a detailed theoretical complexity analysis confirmed with our experimental data.

End Effects in Anti-plane Shear for an Inhomogeneous Isotropic Linearly Elastic Semi-infinite Strip

The purpose of this research is to further investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This is carried out within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. In previous work [1], the elastic coefficients were assumed to be smooth functions of the transverse coordinate so that the material was inhomogeneous in the lateral direction only. Here we develop a new technique, based on a change of variable, to study generally inhomogeneous isotropic materials. The governing partial differential equation is transformed to a Helmholtz equation with a variable coefficient, which facilitates analysis of the influence of material inhomogeneity on the diffusion of end effects. For certain classes of inhomogeneous materials, an explicit optimal decay estimate is established. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials.

Extended Linear Refraction-Diffraction Model

An alternative derivation of the extended refraction-diffraction equation is presented. The bottom curvature and slope-squared terms ignored in the mild-slope equation are included to account for relatively steep and rapidly undulating bathymetry. The linear elliptic partial differential equation is solved by the hybrid element method and applied to three-dimensional bedforms. Quadratic triangular elements are used in the model so that the curvature can be calculated directly from the water depth defined at the nodes. The model is illustrated by simulating wave reflection from a ripple bed and wave transformation over a circular shoal in which previous experimental and numerical results exist. The relative effects of the additional terms and their significance in the model prediction are examined in a numerical experiment.