Self-adjoint Elliptic Equations Research Articles

Asymptotic expansion of the spectral function of elliptic operators inR n

Let {Eλ} be the spectral family of the operator corresponding to a self-adjoint second-adjoint second-order elliptic differential equation or to a first-order elliptic system inR n. The coefficients of the equations are assumed to be constant in a neighborhood of infinity and trapping rays are absent. We obtain the full asymptotic expansion as λ→∞ of the kernel e 1λ (x, y) of the operator dEλ/dλ in terms of rays going from the point y into the point x.

Convergence of SIP

We prove that the Strongly Implicit Procedure (SIP), due to H. L. Stone, for solving selfadjoint elliptic difference equations is convergent.

On the Elliptic Sturmian Theory for General Domains

The well-known comparison theorem by Sturm and Picone for ordinary, self-adjoint, second order differential equations is extended to self-adjoint elliptic differential equations. The basic domain G and the coefficients of the equation are not necessarily bounded, and no regularity hypotheses on the boundary \partial G are required.

Boundary Methods for Solving Elliptic Problems with Singularities and Interfaces

Boundary approximation techniques are described for solving homogeneous self-adjoint elliptic equations. Piecewise expansions into particular solutions are used which approximate both the boundary and interface conditions in a least squares sense. Convergence of such approximations is proved and error estimates are derived in a natural norm. Numerical experiments are reported for the singular Motz problem which yield extremely accurate solutions with only a modest computational effort.

Finite element approximation of the Dirichlet problem using the boundary penalty method

Summary. This paper considers a finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation, A u =f, in a region f2cN" (n=2 or 3) by the boundary penalty method. If the finite element space defined over D h, a union of elements, has approximation power h ~ in the L 2 norm, then (i) for f2=D h convex polyhedral, we show that choosing the penalty parameter e=h x with 2>K yields optimal H 1 and L 2 error bounds if u~HK+I(Q);

Solving Elliptic Difference Equations on a Linear Array of Processors

In this paper we consider the organization of three iterative methods for solving self-adjoint elliptic difference equations on a set of linearly connected processors. These algorithms are the cyclic Chebyshev semi-iterative scheme, a preconditioned conjugate gradient method, and a generalization of the Chebyshev method. We also compare their performance on this multiprocessor as a function of the cost of interprocessor communication.

The Performance of the Collocation and Galerkin Methods with Hermite Bi-Cubics

This paper presents a study of the performance of the collocation and Galerkin methods using Hermite bi-cubic basis functions. It is a sequel to the studies of Houstis et al. [6] and Weiser et al. [15]. The two methods have the linear systems solved by direct methods, band Gauss elimination or Cholesky factorization. The problem domain consists of linear, self-adjoint elliptic equations on two-dimensional rectangular domains. The measures of performance are computer time and memory needed to achieve moderate accuracy. The earlier study comparing finite element and finite difference methods observes that collocation uses less computer time than Galerkin. The second study gave detailed operation counts which support this observation, but also gave substantial experimental evidence to the contrary. We use a new implementation of the collocation method by E. N. Houstis which is tailored for rectangular domains (the one used in Houstis et al. [6] was designed for general domains). We use the same Galerkin implementation that Weiser et al. used. We outline the process of comparing the performance of PDE software and discuss the difficulty of reaching definite conclusions. We analyze the question of error measurement and note that the example given in Weiser et al. [15] as a counterexample to the practice of measuring the error at the grid points or knots (as done in Houstis et al. [6]) is a singular problem and also a counterexample to the approach of measuring the error on a fixed set of points. This study strongly supports the hypothesis that (with these implementations of the methods) collocation performs better than Galerkin for computer time and versus error. We observe that a feature of the particular computing environment causes the Galerkin program to use twice as much memory as necessary and a “normal” implementation should be expected to use nearly the same memory as the collocation program.

Alternating-Direction Incomplete Factorizations

To solve the system of linear equations $Aw = r$ that arises from the discretization of a two-dimensional selfadjoint elliptic differential equation, iterative methods employing easily computed incomplete factorizations, $LU = A + B$, are frequently used. Dupont, Kendall and Rachford [SIAM J. Numer. Anal., 5 (1968), pp. 559–573] showed that, for the DKR factorization, the number of iterations (arithmetic operations) required to reduce the A -norm of the error by a factor of $\varepsilon $ is $O(h^{ - {1 / 2}} \log ({1 / \varepsilon }))(O(h^{ - {5 / 2}} \log ({1 / \varepsilon })))$, where h is the stepsize used in the discretization. We present some error estimates which suggest that, if a pair of alternating-direction DKR factorizations are used, then the number of iterations (arithmetic operations) may be decreased to $O(h^{ - {1 / 3}} \log ({1 / \varepsilon }))(O(h^{ - {7 / 3}} \log ({1 / \varepsilon })))$. Numerical results supporting this estimate are included.

Spectral multigrid methods for elliptic equations

This paper examines the multigrid procedures applied to the iterative solution of spectral equations. Spectral multigrid methods are described for selfadjoint elliptic equations with either periodic or Dirichlet boundary conditions. These methods show a substantial improvement over the simplest iterative schemes.

A discontinuous Steklov problem with an application to water waves

A discontinuous Steklov problem associated with second order selfadjoint elliptic equations which arise in water wave problems is discussed. Here it is assumed that the eigenvalue equation is satisfied only on a part of the boundary. Eigenfunction expansions are derived in an elementary way using a suitable Green's functions. Also a minimax characterization of the eigenvalues is given.

Improperly Posed Initial Value Problems for Self-Adjoint Hyperbolic and Elliptic Equations

Integral representations are obtained for the solution to Cauchy’s problem for hyperbolic equations along a convex time-like surface, the exterior characteristic initial value problem for hyperbolic equations, and Cauchy’s problem for elliptic equations along an analytic surface. Each of these problems is improperly posed in the real domain and hence our representations are constructed by integrating over appropriate regions in the space of one and several complex variables.

On the order of convergence of the Rayleigh-Ritz method with piecewise linear trial functions

COURANT has suggested in [1] a finite difference method which is applicable to the Dirichlet problem for second order self-adjoint elliptic equations in a bounded open two-dimensional region R. The finite difference equations are obtained by means of the Rayleigh-Ritz method with trial functions that are piecewise linear over a triangulation of the plane. This method has been investigated recently by many authors (see [2]-[9]). In the present paper we shall obtain error estimates for the Rayleigh-Ritz method with piecewise linear trial functions under weak regularity assumptions on the boundary of R and the triangulation of the plane of R.

An Approximate Factorization Procedure for Solving Self-Adjoint Elliptic Difference Equations

An Approximate Factorization Procedure for Solving Self-Adjoint Elliptic Difference Equations

A strong comparison theorem for selfadjoint elliptic equations

The purpose of this note is to give a concise proof of a comparison theorem for selfadjoint, second order elliptic equations which yields stronger results than those previously derived in [1], [2] and [3]. All coefficients and domains are to be sufficiently smooth so that the variational techniques of Courant [4] can be applied. Specifically, it is assumed that the first eigenfunction of the selfadjoint boundary value problem

On the convergence of difference schemes approximating the second and third boundary value problems for elliptic equations

THE present paper considers difference schemes approximating the second and third boundary value problems for a self-adjoint elliptic equation without mixed derivatives in a rectangular region, and represents a direct continuation of [1].

A comparison theorem for elliptic differential equations

Clark and the author [2] recently obtained a generalization of the Hartman-Wintner comparison theorem [4] for a pair of self-adjoint second order linear elliptic differential equations. The purpose of this note is to extend this generalization to general second order linear elliptic equations. As in [2], the usual pointwise inequalities for the coefficients are replaced by a more general integral inequality. The result is new even in the one-dimensional case, and extends Leighton's result for self-adjoint ordinary equations [5]. Protter [6] obtained pointwise inequalities in the nonself-adjoint case in two dimensions by the method of Hartman and Wintner [4]. We obtain an alternative to Protter's result as a corollary of our main theorem. Let R be a bounded domain in n-dimensional Euclidean space with boundary B having a piecewise continuous unit normal. The linear elliptic differential operator L defined by

Comparison theorems for elliptic differential equations

In recent years the classical Sturm-Picone comparison theorem for a pair of second order ordinary differential equations has been generalized in two directions. First, Hartman and Wintner [3] have extended the theorem to self-adjoint elliptic equations in n dimensions. On the other hand, Leighton [5] has shown in the case n = 1 that the usual pointwise inequalities for the coefficients can be replaced by a more general inequality. Our main purpose in this note is to extend Leighton's result to self-adjoint elliptic equations. The HartmanWintner result is a corollary of our main theorem. We also give an example to show that our result is actually stronger. An interesting feature is the simplicity of our proof as compared to that of Hartman and Wintner or the recent proof of Kreith [4]. Our result, like Leighton's, is an easy consequence of a theorem from the calculus of variations. A simple proof of the latter theorem in the form required is outlined below; it depends only on Green's formula and an elementary identity. Let R be a bounded, open set in n-dimensional Euclidean space En, with boundary B having a piecewise continuous unit normal. A typical point of En will be denoted by x = (xl, x2, * * *, x8). Partial differentiation with respect to xi will be denoted by Di (i = 1, 2, * * , n). The linear, elliptic self -adjoint partial differential operator L defined by