Linear Elliptic Partial Differential Equations Research Articles

Report of the GDR academy of science

A holomorphic solution for the Riemann-Hilbert boundary value problem in the Sobolev space Wl, pp(D), 2 <p< ∞, is cnnstrueted. This solution satisfies furthermore prescribed jump conditions on a given finite numbcr of simple smooth non-intersecting closed curves inside the domain D of finite measure. The results are then used to solve a similar prohlem for the more general (linear and nonlinearj complex elliptic partial differential equation .

Alternating Direction Collocation for Separable Elliptic Partial Differential Equations

A complete theory is given for the convergence of Alternating Direction Collocation (ADC) applied to a large class of linear separable elliptic partial differential equations on a rectangular domain. It is shown that ADC using bicubic Hermite splines converges in this setting, and converges to the full collocation solution. Moreover, it is shown that a complex embedding of the problem is essential to the theory and that the associated eigensplines and generalized eigensplines can be complex with complex eigenvalues whose geometric and algebraic multiplicities need not be equal. Nonetheless, only real computations are required.

The maximum principle at infinity for minimal surfaces in flat three manifolds

Maximum principles are used as basic analytic tools for studying properties of functions defined on domains in R n and satisfying certain equations (e.g. elliptic). In general these maximum principles play a fundamental role in analysis on complete Riemannian manifolds, especially in the study of variational problems. For example, the well-known maximum principle for harmonic functions has had both a simplifying and unifying effect on the fields of harmonic and complex analysis. H. Hopf [18] gave an important general maximum principle for second order linear elliptic partial differential equations. The Hopf maximum principle easily yields a maximum principle for solutions of the minimal surface equation. In this context the principle states that if D c R 2 is a smooth connected domain and f~ ,f2 are two smooth functions on D that satisfy the minimal surface equation, then the difference fl -f2 cannot have an interior maximum or minimum unless the difference is constant. The maximum principle for minimal graphs gives rise to the following geometric result for minimal surfaces in Riemannian three-manifolds: IfM~ and ME are minimal surfaces in a Riemannian three-manifold that intersect at a common interior point p and M1 is on one side of M2 near p, then M~ intersects M2 in an open surface containing

Matrix-dependent prolongations and restrictions in a blackbox multigrid solver

Multigrid methods are studied for the solution of linear systems resulting from the 9-point discretization of a general linear second-order elliptic partial differential equation in two dimensions. The rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous, or when dominating first-order terms are present. These difficulties may be overcome by choosing the prolongation and restriction operators in a special way. A novel way to do this is proposed. As a result, a blackbox solver (written in standard FORTRAN 77) has been developed. Numerical experiments for several hard test problems are described and comparison is made with other algorithms: the standard MG method and a method introduced by Kettler. A significant improvement of robustness and efficiency is found.

Discretization and solution of elliptic PDEs-a digital signal processing approach

A digital signal processing (DSP) approach is used to study numerical methods for discretizing and solving linear elliptic partial differential equations (PDEs). Whereas conventional PDE analysis techniques rely on matrix analysis and on a space-domain point of view to study the performance of solution methods, the DSP approach described here relies on frequency-domain analysis and on multidimensional DSP techniques. Both discretization schemes and solution methods are discussed. In the area of discretization, mode-dependent finite-difference schemes for general second-order elliptic PDEs are examined, and are illustrated by considering the Poisson, Helmholtz, and convection-diffusion equations as examples. In the area of solution methods, the authors focus on methods applicable to self-adjoint positive definite elliptic PDEs. Both direct and iterative methods are discussed, including fast Poisson solvers, elementary and accelerated relaxation methods, multigrid methods, preconditioned conjugate gradient methods and domain-decomposition techniques. In addition to describing these methods in a DSP setting, an up-to-date survey of recent developments is also provided.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Local existence and regularity of the solution of the nonlinear thin shell model of donnell-mushtari-vlasov

We show the local existence and the regularity of the nonlinear thin shell model of Donnell-Mushtari-Vlasov, using the implicit function theorem at the neighbourhood of the origin and the classical results of regularity for linear elliptic partial differential equations and sytems, for the case of a clamped shell

Mudpack: Multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations

mudpack is a package of portable fortran subprograms which use multigrid iteration for solving real or complex elliptic partial differential equations. The solution regions are rectangles in two dimensions and boxes in three dimensions. Any combination of periodic, Dirichlet, and mixed-derivative boundary conditions is allowed. The equations are automatically discretized using second-order finite differencing. The package will vectorize on Cray computers. Examples are given which demonstrate ease of use, efficiency, and applicability to a wide range of problems.

Eigenvalue estimates and existence for a generalised Graetz problem

The study of forced convection of heat through tubes leads to the study of eigenvalue problems for the Laplacian in which the eigenvalue appears quadratically. Estimates for the eigenvalues based on the eigenvalues of the Laplacian operator are derived. From these estimates the existence of a countable set of eigenvalues is derived. Furthermore, counting multiplicities, the set of eigenvalues can be put in one-to-one correspondence with the eigenvalues of the Laplacian. These results are then generalised to cover a class of selfadjoint linear elliptic partial differential equations in which the eigenvalue appears non-linearly.

Equazioni ellittiche del secondo ordine di tipo non variazionale in aperti non limitati

In this paper we study the Dirichlet problem, for second order, linear elliptic partial differential equations with discontinuous coefficients in unbounded domains. We obtain some results about existence and uniqueness of the solution in W2(Ω).

An Efficient Linear Scheme to Approximate Parabolic Free Boundary Problems: Error Estimates and Implementation

This paper deals with a fully discrete scheme to approximate multidimensional singular parabolic problems; two-phase Stefan problems and porous medium equations are included. The algorithm consists of approximating at each time step a linear elliptic partial differential equation by piecewise linear finite elements and then making an element-by-element algebraic correction to account for the nonlinearity. Several energy error estimates are derived for the physical unknowns; a sharp rate of convergence of $O({h^{1/2}})$ is our main result. The crucial point in implementing the scheme is the efficient resolution of linear systems involved. This topic is discussed, and the results of several numerical experiments are shown.

Singular solutions to some non linear oblique derivative problems

We study some non linear and non elliptic boundary value problems obtained by considering a non linear elliptic second order partial differential equation and a non linear first order boundary condition. Relying on well-known linear results and using the Nash-Moser-type technique of a generalized implicit function theorem of Zehnder, we construct local solutions which are not smooth up to the boundary. Consequently well-known results about the non elliptic linear oblique derivative problem are extended to the non linear case.

An efficient linear scheme to approximate parabolic free boundary problems: error estimates and implementation

This paper deals with a fully discrete scheme to approximate multidimensional singular parabolic problems; two-phase Stefan problems and porous medium equations are included. The algorithm consists of approximating at each time step a linear elliptic partial differential equation by piecewise linear finite elements and then making an element-by-element algebraic correction to account for the nonlinearity. Several energy error estimates are derived for the physical unknowns; a sharp rate of convergence of O ( h 1 / 2 ) O({h^{1/2}}) is our main result. The crucial point in implementing the scheme is the efficient resolution of linear systems involved. This topic is discussed, and the results of several numerical experiments are shown.

A finite difference procedure for solving coupled, nonlinear elliptic partial differential equations

A finite difference procedure is presented for solving coupled sets of partial differential equations. For one dependent variable, the procedure consists of replacing the concept of a single unknown at multiple grid points with the concept of a line of node points with multiple unknowns at each node point. The procedure is illustrated first for a second order, linear elliptic partial differential equation and then for a coupled set of non-linear elliptic partial differential equations. The method is easier to use and requires less computer storage than a banded solver method such as IMSL's routine LEQT1B. The procedure could be extended to include three spatial coordinates and time.

Numerical simulation of electrodeposition problems

AbstractThe mathematical modelling of the electrodeposition phenomenon leads to a linear elliptic partial differential equation subject to non‐linear mixed boundary contitions. This is complicated further by the fact that the domain under consideration is evolving with time.Owing to the nature of the mathematical model, the boundary element method is a very efficient numerical technique for the solution of such problems. The authors report their numerical simulation of the above problem which also has applications in the cathodic protection systems for corrosion prevention.

Evaluation of the residual vector in global element calculations

The defining equations for the global element method [5], applied to the solution of (linear or nonlinear) elliptic partial differential equations, are most efficiently solved using an iterative scheme. It has been shown previously [6] that for a calculation in two dimensions, the operation count of such a scheme is O ( MN 4) where M is the number of elements and N the size of the expansion used to approximate the solution in each element. This operation count results partly from the need to multiply an MN 2 vector by an MN 2 × MN 2 matrix to form the residual vector at each iteration. We demonstrate here how the residual can be computed with the improved operation count of O ( MN 2 ln N): an additional advantage of the new scheme is that the full diagonal blocks of the coefficient matrix need not be assembled.

A feedback finite element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator

This paper is the first in a series of two in which we discuss some theoretical and practical aspects of a feedback finite element method for solving systems of linear second-order elliptic partial differential equations (with particular interest in classical linear elasticity). In this first part we introduce some nonstandard finite element spaces, which, though based on the usual square bilinear elements, permit local mesh refinement. The algebraic structure of these spaces and their approximation properties are analyzed. An “equivalent estimator” for the H 1 finite element error is developed. In the second paper we shall discuss the asymptotic properties of the estimator and computational experience.

Tensor Product Generalized ADI Methods for Separable Elliptic Problems

We consider solving separable, second order, linear elliptic partial differential equations. If an elliptic problem is separable, then, for certain discretizations, the matrices involved in the corresponding discrete problem can be expressed in terms of tensor products of lower order matrices. In the most general case, the discrete problem can be written in the form $(A_1 \otimes B_2 + B_1 \otimes A_2 )C = F$. We present a new Tensor Product Generalized Alternating Direction Implicit (TPGADI) iterative method for solving such discrete problems. We prove convergence and establish computational efficiency. The TPGADI method is applied to the Hermite bicubic collocation equations. We conclude that the TPGADI method is an effective tool for solving the discrete elliptic problems arising from a large class of elliptic problems.

A tensor product generalized ADI method for elliptic problems on cylindrical domains with holes

We consider solving second order linear elliptic partial differential equations together with Dirichlet boundary conditions in three dimensions on cylindrical domains (nonrectangular in x and y) with holes. We approximate the partial differential operators by standard partial difference operators. If the partial differential operator separates into two terms, one depending on x and y, and one depending on z, then the discrete elliptic problem may be written in tensor product form as ( T z ⊗ I + I⊗ A xy ) U= F. We consider a specific implementation which uses a Method of Planes approach with unequally spaced finite differences in the xy direction and symmetric finite difference in the z direction. We establish the convergence of the Tensor Product Generalized Alternating Direction Implicit iterative method applied to such discrete problems. We show that this method gives a fast and memory efficient scheme for solving a large class of elliptic problems.

A boundary element method for the solution of the transient diffusion equation in two dimensions

The main advantage of using potential or boundary element methods for generating numerical solutions to linear elliptic partial differential equations is that the dimension of the problem is reduced by one. This feature not only reduces the number of algebraic equations to be solved, but, in two and three space dimensions, it also eliminates the problems of mesh generation. Both of these factors contribute to reducing the computational cost. A potential method is described which provides numerical solutions of the transient diffusion equation for two-dimensional problems. The solutions are generally as accurate as conventional techniques whilst retaining the advantage of a reduction in dimensionality throughout the time domain. The numerical study, however, also indicates that the method may have limitations, particularly when there are discontinuities at the boundaries.

Uniqueness of the dirichlet problem for a linear elliptic partial differential equation of the second order

On donne un theoreme d'unicite pour les solutions regulieres du probleme de Dirichlet associe a une equation aux derivees partielles lineaire uniformement elliptique d'ordre 2 dans un domaine exterieur regulier Ω de R n