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Displaying 21 – 40 of 68

Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle

Nachman Aronszajn, R. D. Brown, R. S. Butcher (1973)

Annales de l'institut Fourier

A technique is developed for constructing the solution of $Δ^{2} u = F$ in $R = {(x, y) : | x | < a, | y | < b}$ , subject to boundary conditions $u = φ$ , $\frac{\partial u}{\partial n} = ψ$ on $\partial R$ . The problem is reduced to that of finding the orthogonal projection $P w$ of $w$ in $L^{2} (R)$ onto the subspace $H$ of square integrable functions harmonic in $R$ . This problem is solved by decomposition $H$ into the closed direct (not orthogonal) sum of two subspaces $H^{(1)}, H^{(2)}$ for which complete orthogonal bases are known. $P$ is expressed in terms of the projections $P^{(1)}$ , $P^{(2)}$ of $L^{2} (R)$ onto $H^{(1)}$ , $H^{(2)}$ respectively. The resulting construction...

Convergence of $V$ -cycle and $F$ -cycle multigrid methods for the biharmonic problem using the Morley element.

Zhao, Jie (2004)

ETNA. Electronic Transactions on Numerical Analysis [electronic only]

Dirichlet Finite Biharmonic Functions with Dirichlet Finite Laplacians.

Mitsuru Nakai, Leo Sario (1971)

Mathematische Zeitschrift

Dirichlet problems for the biharmonic equation.

Begehr, Heinrich (2005)

General Mathematics

Duality in the obstacle and unilateral problem for the biharmonic operator

Ján Lovíšek (1981)

Aplikace matematiky

The paper presents a problem of duality for the obstacle and unilateral biharmonic problem (the equilibrium of a thin plate with an obstacle inside the domain or on the boundary). The dual variational inequality is derived by introducing polar functions.

Error estimates for mixed methods

R. S. Falk, J. E. Osborn (1980)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Error estimates for the assumed stresses hybrid methods in the approximation of 4th order elliptic equations

Alfio Quarteroni (1979)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Existence of bounded biharmonic functions.

Mitsuru Nakai, Leo Sario (1973)

Journal für die reine und angewandte Mathematik

Existence of solutions to the Poisson equation in $L_{p}$ -weighted spaces

Joanna Rencławowicz, Wojciech M. Zajączkowski (2010)

Applicationes Mathematicae

We examine the Poisson equation with boundary conditions on a cylinder in a weighted space of $L_{p}$ , p≥ 3, type. The weight is a positive power of the distance from a distinguished plane. To prove the existence of solutions we use our result on existence in a weighted L₂ space.

Existence of solutions to the Poisson equation in L₂-weighted spaces

Joanna Rencławowicz, Wojciech M. Zajączkowski (2010)

Applicationes Mathematicae

We consider the Poisson equation with the Dirichlet and the Neumann boundary conditions in weighted Sobolev spaces. The weight is a positive power of the distance to a distinguished plane. We prove the existence of solutions in a suitably defined weighted space.

Existence of the boundary value of a polyharmonic function.

Mikhailov, V.P. (2005)

Sibirskij Matematicheskij Zhurnal

Fast Poisson Solvers on General Two Dimensional Regions for the Dirichlet Problem.

A.S.L. Shieh (1978/1979)

Numerische Mathematik

Fast solution of periodic integral equations by GMRES.

Plato, R., Vainikko, G. (2001)

Computational Methods in Applied Mathematics

Functions of a quaternion variable which are gradients of real-valued functions.

R.R. Kocherlakota (1986)

Aequationes mathematicae

Generators of the Space of Bounded Biharmonic Functions.

Leo Sario, Cecilia Wang (1972)

Mathematische Zeitschrift

Internal finite element approximation in the dual variational method for the biharmonic problem

Ivan Hlaváček, Michal Křížek (1985)

Aplikace matematiky

A conformal finite element method is investigated for a dual variational formulation of the biharmonic problem with mixed boundary conditions on domains with piecewise smooth curved boundary. Thus in the problem of elastic plate the bending moments are calculated directly. For the construction of finite elements a vector potential is used together with $C^{0}$ -elements. The convergence of the method is proved and an algorithm described.

Liouville theorem for Dunkl polyharmonic functions.

Ren, Guangbin, Liu, Liang (2008)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Method of fundamental solutions for biharmonic equation based on Almansi-type decomposition

Koya Sakakibara (2017)

Applications of Mathematics

The aim of this paper is to analyze mathematically the method of fundamental solutions applied to the biharmonic problem. The key idea is to use Almansi-type decomposition of biharmonic functions, which enables us to represent the biharmonic function in terms of two harmonic functions. Based on this decomposition, we prove that an approximate solution exists uniquely and that the approximation error decays exponentially with respect to the number of the singular points. We finally present results...

Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates

D. N. Arnold, F. Brezzi (1985)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Multiple layer potentials for the quadrant and their application to the Dirichlet problem in plane domains with a piecewise smooth boundary

Günter Albinus (1983)

Banach Center Publications

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