Showing papers on "Free boundary problem published in 1977"

Multi-level adaptive solutions to boundary-value problems

Abstract: The boundary-value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes. Interactions between these levels enable us (i) to solve the possibly nonlinear system of n discrete equations in 0(n) operations (40n additions and shifts for Poisson problems); (ii) to conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining \"°°-order\" approximations and low n, even when singularities are present. General theoretical analysis of the numerical process. Numerical experiments with linear and nonlinear, elliptic and mixed-type (transonic flow) problemsconfirm theoretical predictions. Similar techniques for initial-value problems are briefly

Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions

Abstract: In this paper it is shown that the analysis of Titchmarsh's book [32] for regular Sturm-Liouville problems on a finite closed interval carries over readily to regular problems involving the eigenvalue parameter in the boundary condition at one end-point. The manner in which this type of problem is associated with a self-adjoint operator in Hilbert space has recently been pointed out by Walter in [36], and his operator-theoretic formulation is adopted here. The use of the eigenfunction expansion is illustrated by applying it to solve a heat-conduction problem for a solid in contact with a fluid.

The regularity of free boundaries in higher dimensions

Abstract: The problem of studying the regularity of the free boundary that arises when considering the energy minimizing function over the set of those functions bigger than a given "obstacle" has been the subject of intensive research in the last decade. Let me mention H. Lewy and G. Stampacchia [14], D. Kinderlehrer [11], J. C. Nitsche [15] and N. M. Riviere and the author [5] among others. In two dimensions, by the use of analytic reflection techniques due mainly to H. Lewy [13], much was achieved. Recently, the author was able to prove, in a three dimensional filtration problem [4], that the resulting free surface is of class C 1 and all the second derivatives of the variational solution are continuous up to the free boundary, on the non-coincidence set. This fact has not only the virtue of proving that the variational solution is a classical one, but also verifies the hypothesis necessary to apply a recent result due to D. Kinderlehrer and L. Nirenberg, [12] to conclude that the free boundary is as smooth as the obstacle. Nevertheless, in that paper ([4]), strong use was made of the geometry of the problem: this implied that the free boundary was Lipschitz. Also it was apparently essential that the Laplacian of the obstacle was constant. In the first part of this paper we plan to treat the general non-linear free boundary problem as presented in H. Brezis-D. Kinderlehrer [2]. Our main purpose is to prove that if X 0 is a point of density for the coincidence set, in a neighborhood of X 0 the free boundary is a C 1 surface and all the second derivatives of the solution are continuous up to it. In the second part we will s tudy the parabolic case (one phase Stefan problem) as presented by G. Duvaut [7] or A. Friedman and D. Kinderlehrer [9]. There we prove that if for a fixed time, to, the point X 0 is a density point for the coincidence set (the ice) then in a

Regularity in free boundary problems

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The Propagation of Singularities along Gliding Rays.

Abstract: LetP be a second-order differential operator with real principal symbol and fibre-simple characteristics on a manifold with boundary non-characteristic forP. LetB be a differential operator such that the boundary value problem (P, B) is normal and satisfies the Lopatinskii-Schapiro condition. The singularities of distributions,u, such thatP u is smooth on the boundary, near points at which the boundary is bicharacteristically convex are shown to propagate, in the boundary, only along the gliding rays, which are the leaves of the Hamilton foliation of the glancing surface. This analysis, combined with known results on diffraction, leads to a Poisson relation bounding the singular support of the Fourier transform of the Dirichlet spectral density for a compact Riemannian manifold with geodesically convex, or concave, boundary in terms of the geodesic length spectrum.

Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains.

Abstract: A general boundary value problem of variational type is considered for a general quasi-linear elliptic partial differential operator of order 2m in generalized divergence form. Such problems are considered on an arbitrary domain in a Euclidean space without hypotheses of boundedness on the domain or smoothness on its boundary. Contrary to the prevailing doctrine in the literature, it is shown that the corresponding operator between Banach spaces is pseudo-monotone, and that a wide variety of existence results can be derived from this fact.

On the solution of large, structured linear complementarity problems: The block partitioned case

Abstract: This paper delineates the underlying theory of an efficient method for solving a class of specially-structured linear complementarity problems of potentially very large size. We propose a block-iterative method in which the subproblems are linear complementarity problems. Problems of the type considered here arise in the process of making discrete approximations to differential equations in the presence of special side conditions. This problem source is exemplified by the free boundary problem for finite-length journal bearings. Some of the authors' computational experience with the method is presented.

Boundary regularlty for solutions to various capillarity and free boundary problems

Abstract: (1977). Boundary regularlty for solutions to various capillarity and free boundary problems. Communications in Partial Differential Equations: Vol. 2, No. 4, pp. 323-357.

A variational approach to singularly perturbed boundary value problems for ordinary and partial differential equations with turning points

Abstract: In studying singularity perturbed boundary value problems for second order linear differential equations with a simple turning point, R. C. Ackerberg and R. E. O’Malley [2] pointed out a number of interesting anomalies. In particular they observed that standard application of the method of matched asymptotic expansions did not suffice to uniquely determine the asymptotic expansion of the solution. They further noted that the standard construction in that method led to boundary layers at both ends of the interval, even for problems where in fact there is only one boundary layer located at one or other of the endpoints. In this paper we employ a variational formulation of the problem to resolve the question of the number and location of the boundary layers as well as to uniquely determine the asymptotic expansion of the solution. The results are then extended to analogous problems for partial differential equations, and new results are obtained for a class of singularly perturbed elliptic boundary value pro...

The fluid flow through porous media. Regularity of the free surface

Abstract: The fluid flow through an earth dam separating two water reservoirs of different levels gives rise to a free boundary problem. In [1] we have proved the existence of a solution to this problem. In this paper we show that the free boundary is regular.

Constructing orthogonal curvilinear meshes by solving initial value problems

Abstract: The purpose of this paper is to develop methods for constructing orthogonal curvilinear meshes suitable for solving partial differential equations over plane regions with smooth, curved boundaries. These curved meshes cover an annular strip along the boundary of the region which is included in the mesh. In this strip difference approximations of partial differential equations and boundary conditions can be set up as easily as they can for halfspace problems. The rest of the region and a suitable part of the annular strip can be covered by a square or rectangular mesh. In the present paper we consider the problem of determining curved meshes by solving nonlinear hyperbolic initial value problems which are formally related to the Cauchy-Riemann equations.

One-Dimensional Parabolic Free Boundary Problems

Abstract: Frequently, diffusion processes require the determination of a free surface from overprescribed boundary data. A commonly used constructive solution technique for such problems is the method of straight lines. This paper illustrates the steps involved in the solution process. Specifically, the method of lines is used to approximate various explicit and implicit free boundary problems for a linear one-dimensional diffusion equation with a sequence of free boundary problems for ordinary differential equations. It is shown that these equations have solutions which can be readily obtained with the method of invariant imbedding. It also is established for a model problem that the approximate solutions converge to a unique (almost) classical solution as the discretization parameter goes to zero.

A minimum principle for the principal eigenvalue for second-order linear elliptic equations with natural boundary conditions

Abstract: : This paper gives a new characterization of the smallest eigenvalue for second order linear elliptic partial differential equations, not necessarily self-adjoint, with both natural and Dirichlet boundary conditions, and also give a new alternative numerical method for calculating both the smallest eigenvalue and corresponding eigenvector in the case of natural boundary conditions. The smallest eigenvalue, if appropriate sign changes are made, determines the stability of equilibrium solutions to certain second order nonlinear partial differential equations. The corresponding eigenvector enables one to determine the first approximation of the solution of the nonlinear equation to variations of the initial conditions from the equilibrium solution. These nonlinear equations are important in the applications. For these reasons it is important to have these characterizations of the smallest eigenvalue and eigenvector. Our method converts the determination of the eigenvalue and eigenvector to determining the solution of a stationary stochastic control problem. This latter problem is solved and from it a numerical scheme arises naturally. This method appears to have applications in solving other problems.

A classification and survey of numerical methods for boundary value problems in ordinary differential equations

Abstract: The objective of the paper is a survey and classification of available numerical methods for boundary value problems with an emphasis on two point boundary value problems. In the paper a general two point boundary value problem is stated first and later available numerical methods are classified into five main groups according to their approach in numerical solution. In the next sections of the paper, each of these main groups are described and appropriate subgroups are defined and discussed.