Solution Of Dirichlet Research Articles

On the direct solution of Dirichlet's problem in two dimensions

General methods for the solution of the two-dimensionalDirichlet problem are presented. They are general in the sense that they are independent of the numerical values of the entries of the coefficient matrix. The algorithms proposed utilize the band structure of that matrix, to increase computational efficiency. Both direct inversion and factorization of the coefficient matrix are dealt with. The algorithms are tailored for using backing storage but saving main computer storage space.

Pointwise Bounds for Discrete Green’s Functions

Previous article Next article Pointwise Bounds for Discrete Green’s FunctionsJ. H. Bramble and V. ThoméeJ. H. Bramble and V. Thoméehttps://doi.org/10.1137/0706053PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. H. Bramble, On the convergence of difference approximations to weak solutions of Dirichlet's problem, Numer. Math., 13 (1969), 101–111 10.1007/BF02163229 MR0250495 0196.17803 CrossrefISIGoogle Scholar[2] James H. Bramble, On the convergence of difference approximations for second order uniformly elliptic operatorsNumerical Solution of Field Problems in Continuum Physics (Proc. Sympos. Appl. Math., Durham, N.C., 1968), SIAM-AMS Proc., Vol. II, Amer. Math. Soc., Providence, R. I., 1970, 201–209 MR0260200 0234.65086 Google Scholar[3] J. H. Bramble, , B. E. Hubbard and , Vidar Thomée, Convergence estimates for essentially positive type discrete Dirichlet problems, Math. Comp., 23 (1969), 695–709 MR0266444 0217.21902 CrossrefISIGoogle Scholar[4] J. H. Bramble, , B. E. Hubbard and , M. Zlámal, Discrete analogues of the Dirichlet problem with isolated singularities, SIAM J. Numer. Anal., 5 (1968), 1–25 10.1137/0705001 MR0239770 0176.46902 LinkISIGoogle Scholar[5] James R. Kuttler, Finite difference approximations for eigenvalues of uniformly elliptic operators, SIAM J. Numer. Anal., 7 (1970), 206–232 10.1137/0707014 MR0273841 0208.42601 LinkISIGoogle Scholar[6] Pentti Laasonen, On the solution of Poisson's difference equation, J. Assoc. Comput. Mach., 5 (1958), 370–382 MR0121999 0087.12202 CrossrefISIGoogle Scholar[7] Vidar Thomée and , Bertil Westergren, Elliptic difference equations and interior regularity, Numer. Math., 11 (1968), 196–210 MR0224303 0159.38204 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Asymptotics of the heat kernels on 2D latticesAsymptotic Analysis, Vol. 112, No. 1-2 | 6 Mar 2019 Cross Ref On Solving the Singular System Arisen from Poisson Equation with Neumann Boundary ConditionJournal of Scientific Computing, Vol. 69, No. 1 | 4 April 2016 Cross Ref Petrov–Galerkin finite element approximation for the three dimensional well modelJournal of Computational and Applied Mathematics, Vol. 277 | 1 Mar 2015 Cross Ref A 3-dimensional well model in the flow transport through porous mediaApplied Mathematical Modelling, Vol. 38, No. 21-22 | 1 Nov 2014 Cross Ref On the decay of the inverse of matrices that are sum of Kronecker productsLinear Algebra and its Applications, Vol. 452 | 1 Jul 2014 Cross Ref On Sojourn Times in the M/ M/1-PS Model, Conditioned on the Number of Other UsersApplied Mathematics Research eXpress, Vol. 11 | 19 February 2010 Cross Ref A kernel-free boundary integral method for elliptic boundary value problemsJournal of Computational Physics, Vol. 227, No. 2 | 1 Dec 2007 Cross Ref On the accuracy of finite difference methods for elliptic problems with interfacesCommunications in Applied Mathematics and Computational Science, Vol. 1, No. 1 | 31 December 2006 Cross Ref Least squares methods for elliptic systemsMathematics of Computation, Vol. 44, No. 169 | 1 January 1985 Cross Ref L�sung des Dirichletproblems und Konvergenz der Differenzenapproximation f�r elliptische Differentialgleichungen zweiter OrdnungMathematische Zeitschrift, Vol. 131, No. 3 | 1 Sep 1973 Cross Ref Discrete Green’s functionsMathematics of Computation, Vol. 27, No. 121 | 1 January 1973 Cross Ref The distribution of the eigenvalues of the discrete LaplacianBIT, Vol. 12, No. 4 | 1 Dec 1972 Cross Ref A fourth-order finite-difference approximation for the fixed membrane eigenproblemMathematics of Computation, Vol. 25, No. 114 | 1 January 1971 Cross Ref Volume 6, Issue 4| 1969SIAM Journal on Numerical Analysis523-616 History Submitted:14 November 1968Published online:14 July 2006 InformationCopyright © 1969 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0706053Article page range:pp. 583-590ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

On the convergence of difference approximations to weak solutions of Dirichlet's problem

On the convergence of difference approximations to weak solutions of Dirichlet's problem

Tikhonov's “Regularization Method” for weak solutions of boundary value problems

TIKHONOV [1] has proposed a method for the approximate solution of degenerate systems of linear algebraic equations. This paper contains a generalization of this method to the infinite-dimensional case and an application to the approximate solution of Dirichlet's problem for the wave equation.

Solution of dirichlet's problem for Laplace's equation by the method of statistical tests

BELOW it is shown that the solution of the internal and external Dirichlet problems for Laplace's equation may be obtained by the method of statistical tests if the region of definition of the solution is presented in the form of the superposition of partially intersecting subregions for which the solution is known. According to the given boundary conditions the required solution is defined in terms of probability characteristics of the Markov process of random transitions between points of the boundaries of the subregions. Examples of the practical application of the method of statistical tests in solving Dirichlet problems are given.

Die Lösung des Dirichletschen und Neumannschen Problems für die Poissonsche Gleichung auf Flächen mit Hilfe der Greenschen Funktion

Die Lösung des Dirichletschen und Neumannschen Problems für die Poissonsche Gleichung auf Flächen mit Hilfe der Greenschen Funktion

The properties of an iterational process

THE iterational process for the numerical solution of Dirichlet's problem for Poisson's equation in two independent variables is discussed. (Some of the author's results on this topic were presented to the Third Siberian Conference on Mathematics and Mechanics in 1964 [1]. Though some similar papers have been published (e.g. [2, 3]), the iterational method considered below has several special features and should therefore be of interest.)

The expansion of solutions of dirichlet's problem and a mixed problem for a biharmonic equation in a series of solutions of reducing problems

The expansion of solutions of dirichlet's problem and a mixed problem for a biharmonic equation in a series of solutions of reducing problems

Approximation theorems for the solution of Fourier's problem and Dirichlet's problem

Approximation theorems for the solution of Fourier's problem and Dirichlet's problem

Dirichlet’s Problem for an Equation with Periodic Coefficients Depending on a Small Parameter

This paper studies the limiting behavior of the solution $u^\varepsilon (x)$ of Dirichlet’s problem for \[ L^\varepsilon u^\varepsilon = \frac{1}{2}\sum {a_{ij} } \left( {\frac{x}{\varepsilon }} \right)\frac{{\partial ^2 u^\varepsilon }}{{\partial x^i \partial x^j }} + \sum {b_i } \left( {\frac{x}{\varepsilon }} \right)\frac{{\partial u^\varepsilon }}{{\partial x^i }} - c\left( {\frac{x}{\varepsilon }} \right)u^\varepsilon = 0, \] when $\varepsilon \to 0$. The coefficients of the operator $L^1 $ are assumed to be periodic.It is proved that $\lim _{\varepsilon \to 0} u^\varepsilon (x) = u(x)$ exists. The function $u(x)$ is a solution of Dirichlet’s problem for the equation $\bar Lu = 0$, where the coefficients of the operator $\bar L$ are obtained by averaging the coefficients of the operator $L^\varepsilon $.

Heat-Transfer Coefficient of Inviscid Fluid Freezing Onto a Moving Heat Sink

When a cold slab travels at speed U through a liquid metal bath, it freezes out metal at the rate of V. It is shown that the problem of determining the heat-transfer coefficient h at the interface of the liquid and solid phases is equivalent—under certain simplifying assumptions—to solving the (time-independent) wave equation in a sector. For the case of α ≡ arctan V/U < π/4, treated in the present paper, the problem is reduced, through suitable changes of variables and a Fourier sine transform, to the solution of Dirichlet’s problem for the Laplace equation. The temperature field T(X, Y) is expressed as an inverse sine transform, involving a single integration (in the transform variable). One finds that at the entrance cross section to the bath, X = 0, the heat-transfer coefficient is zero, then it rapidly approaches the asymptotic (“fully developed”) value h = γcV cos α. The heat-transfer coefficient is determined in closed form for α = π/6, π/4, and asymptotic expressions of it are derived for very small and very large distances from the origin when α is arbitrary (0 < α < π/4). Numerical evaluation of the heat-transfer coefficient at the solid-liquid interface is carried out for α = π/36, π/12, π/4. Plots of the temperature field T for these angles are also shown, along rays ϑ = π/6, π/3, π/2 to the horizontal.

The method of potentials and the numerical solution of Dirichlet' s problem for the Laplace equation

The method of potentials and the numerical solution of Dirichlet' s problem for the Laplace equation

On the method of nets for a boundary problem with an oblique and a normal derivative

In this article we construct error bounds for the solution of Poisson's equation by the method of nets, where the oblique or normal derivative is given on the boundary of the region. We shall be considering a system of difference equations, equal in number to the number of nodes of the net lying inside the region less one. The value of the function at one node is fixed and a corresponding equation “rejected”, and the system of difference equations thus reduced has a unique solution which can be calculated by an iterative method. An arbitrary constant is added to the resulting approximate solution. The book [1] contains a practical guide to the application of this method with a very simple approximation to the derivative on the boundary. However, the error bounds for the method in the solution of the problem with a given oblique or normal derivative on the boundary of the region have not been discussed before. The solution of the problem with an oblique derivative can be reduced to the solution of Hilbert's problem using the method of nets in [2], but since it is assumed that six derivatives of the required solution exist, the estimates obtained there are rough, of the order of O( h). The method of nets for Neumann's problem is discussed in [3]–[6]. In [3], in particular, a system of self-conjugate difference equations is constructed for the multidimensional case, its solution converging weakly to the generalized solution of Neumann's problem in the space W 2 (1) as h → 0. The difference scheme for the two-dimensional case is considered by reducing Neumann's problem for the Laplace equation to the Dirichlet problem, and in [5] the corresponding error bounds, of order O( h), are derived, on the assumption that the third derivatives of the required solution in the closed region are continuous and satisfy the Hölder condition, but that the fourth derivatives may have singularities at a finite number of points on the boundary. In the three-dimensional case, the method of orthogonal projections is used in [3] and [4]. In the solution of Neumann's problem, the local approximation errors of this method at internal nodes of the net are of the order of O( h 2). The error bounds of the difference ratios are evaluated in [5], and in [6] the smoothness of the difference solution inside the region is discussed. Finally, we must mention [7], which gives the error bounds of the method of nets for the third boundary problem and for mixed boundary conditions, when the normal derivative is given on part of the boundary. However, these bounds do not apply to Neumann's problem or to the problem with an oblique derivative on the whole boundary for Laplace and Poisson's equations, which we consider here. To approximate to the oblique and normal derivative we shall use the difference operators of high accuracy, which were suggested in [8]–[11], and, assuming that the fourth derivatives of the required solution are bounded, we derive error bounds of order O(h 2−2 ¦in h¦ In 2 h) , where h is the step of the net for the two-dimensional case. If first difference relations are used to define the derivative on the boundary of the region (see [12], [1]). the error estimate has the order O(h 1−1 ¦in h¦ in 2 h) , This is done for the two cases of “starred” boundary conditions (§ 2) and of an oblique derivative whose direction does not coincide with that of the tangent to the boundary (§ 3). In the concluding paragraph (§ 4) we consider the application of boundary difference operators of othe types [3], make estimates for the multidimensional case and give a new method for constructing the error bounds in the solution of Dirichlet's problem for the Laplace equation by the method of nets.

Approximation of solutions of Dirichlet's problem on nearly circular domains and their application in numerical methods

Approximation of solutions of Dirichlet's problem on nearly circular domains and their application in numerical methods

Penetration of a wedge into compressible half-space

This paper presents expressions for the classical combined field integral equations for the solution of Dirichlet and Neumann exterior Helmholtz problems on the plane, in terms of smooth (continuously differentiable) integrands. These expressions are obtained by means of a singularity subtraction technique based on pointwise plane-wave expansions of the unknown density function. In particular, a novel regularization of the hypersingular operator is obtained, which, unlike regularizations based on Maue's integration-by-parts formula, does not give rise to involved Cauchy principal value integrals. Moreover, the expressions for the combined field integral operators and layer potentials presented in this contribution can be numerically evaluated at target points that are arbitrarily close to the boundary without severely compromising their accuracy. A variety of numerical examples in two spatial dimensions that consider three different Nyström discretizations for smooth domains and domains with corners—one of which is based on direct application of the trapezoidal rule—demonstrates the effectiveness of the proposed higher-order singularity subtraction approach.

Functions which have harmonic support

1. Introduction. This paper presents some properties of the functions in a certain subclass of the class of subharmonic functions. Consider real valued functions v(z) defined in a domain D of the complex plane. DEFINITION 1. The function v is in (hs), the class of functions which have harmonic support in D, if and only if v is uniformly bounded on each compact subset of D and for each zoC=Dah harmonic and single valued in D3h(zo) =v(zo) and h<v in D. Caratheodory [5] used such functions in constructing a general result on solutions of Dirichlet's problem which are based on convergent sequences of continuous operators. The functions of the class (hs) are continuous subharmonic functions, but Brelot [4, footnote 16] showed the existence of subharmonic functions in Cn which are not in (hs). A simple example of a continuous subharmonic function not in (hs) is the function v = z a, 0 <a <1, in zI <1. For suppose v C (hs) and let hn denote the support functions at Zn= l/n, n= 2, 3, * - - . The sequence { hn } constitutes a bounded family of harmonic functions which is thus normal. Writing z = x+iy, the family

Asymptotic development of the solution of Dirichlet’s problem at analytic corners

Asymptotic development of the solution of Dirichlet’s problem at analytic corners