Ritz approximations to two-dimensional boundary value problems

Summary Various Ritz solutions to the plane strain elasticity and the steady state heat flow boundary value problems for a polygonal domain are considered. Historically, two basic approaches have been used in partitioning into finite elements, (i) complete triangulation and (ii) rectangles with boundary triangles. In each case, the Ritz solution is the unique function (or vector of functions) which minimizes an energy functional over a finite dimensional vector spaceS. We consider as choices forS, piecewise linear and cubic functions for complete triangulations; and piecewise bilinear and bicubic functions for the case in which is a union of rectangles and boundary triangles. For the elasticity problem,L ₂ convergence of the components of stress and strain is established for each choice of the spaceS. L ₂ convergence of the displacement vector is also shown for a wide class of boundary conditions. Convergence of the temperature is proven for the heat flow problem also. Numerical comparisons are made of the Ritz solutions based upon each spaceS of trial functions.     

The Oblique Derivative Problem for the Laplace Equation in a Plain Domain

The oblique derivative problem for the Laplace equation is studied in a planar multiply connected domain. The boundary condition has a form where is the unit normal vector, is the unit tangential vector and is a fixed real number. If is a Hölderian function and the corresponding domain has Ljapunov boundary then the classical problem is studied. If on the boundary and the domain has a locally Lipschitz boundary then a solution, which fulfils the boundary condition in the sense of a nontangential limit, is studied. If is a real measure on the boundary and the domain has bounded cyclic variation then a solution in a sense of distributions is studied. The solution is looked for in a form of a linear combination of a single layer potential and an angular potential.     

Approximation by cubicC 1-splines on arbitrary triangulations

Summary In this paper, we present an efficient representation for bivariate piecewise cubicC ¹-splines on arbitrary triangulations. A numerical method is discussed for computing the dimension of the spaceS ₃ ¹ () of these splines. We consider subspaces ofS ₃ ¹ () satisfying certain boundary conditions. Some applications are given where piecewise cubicC ¹-functions are used to solve interpolation problems and least squares approximation problems.     

Total flux estimates for a finite element approximation of the Dirichlet problem using the boundary penalty method

Summary This paper considers a fully practical piecewise linear finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation,Au=f, in a smooth region< ⁿ (n=2 or 3) by the boundary penalty method. Using an unfitted mesh; that is ^h , an approximation of with dist (, ^h )Ch ² is not in general a union of elements; and assuminguH ⁴ () we show that one can recover the total flux across a segment of the boundary of with an error ofO(h ²⁾. We use these results to study a fully practical piecewise linear finite element approximation of an elliptic equation by the boundary penalty method when the prescribed data on part of the boundary is the total flux.Supported by a SERC research studentship     

Monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem

Summary In this paper, we consider monotone explicit iterations of the finite element schemes for the nonlinear equations associated with the boundary value problem u=bu ², based on piecewise linear polynomials and the lumping operator. These iterations construct the monotonically decreasing and increasing sequences, and convergence proofs are given. Finally, we present some numerical examples verifying the effectiveness of the theory.     

Midpoint collocation for Cauchy singular integral equations

Summary A Cauchy singular integral equation on a smooth closed curve may be solved numerically using continuous piecewise linear functions and collocation at the midpoints of the underlying grid. Even if the grid is non-uniform, suboptimal rates of convergence are proved using a discrete maximum principle for a modified form of the collocation equations. The same techniques prove negative norm estimates when midpoint collocation is used to determine piecewise constant approximations to the solution of first kind equations with the logarithmic potential.This work was supported by the Australian Research Council through the program grant Numerical analysis for integrals, integral equations and boundary value problems     

Numerische Ermittlung quasikonformer Abbildungen mit finiten Elementen

Summary In this note we consider so called p-analytic mappings of simply or doubly connected domains on rectangles or circular rings. Real and imaginary parts of the mappings can be described by minimal-principles. By minimizing the corresponding functionals in a class of linear or bilinear finite elements we obtain an approximation of the mapping and also upper and lower bounds for the p-module of a domain with polygonal boundary. Error bounds are given for smooth and for piecewise constant functionsp. We present numerical experiments.

     

On Korn's first inequality for tangential or normal boundary conditions with explicit constants

We will prove that for piecewise C²‐concave domains in Korn's first inequality holds for vector fields satisfying homogeneous normal or tangential boundary conditions with explicit Korn constant . Copyright © 2016 John Wiley & Sons, Ltd.     

Second-order finite element approximations and a posteriori error estimation for two-dimensional parabolic systems

Summary We discuss an adaptive local refinement finite element method of lines for solving vector systems of parabolic partial differential equations on two-dimensional rectangular regions. The partial differential system is discretized in space using a Galerkin approach with piecewise eight-node serendipity functions. An a posteriori estimate of the spatial discretization error of the finite element solution is obtained using piecewise fifth degree polynomials that vanish on the edges of the rectangular elements of a grid. Ordinary differential equations for the finite element solution and error estimate are integrated in time using software for stiff differential systems. The error estimate is used to control a local spatial mesh refinement procedure in an attempt to keep a global measure of the error within prescribed limits. Examples appraising the accuracy of the solution and error estimate and the computational efficiency of the procedure relative to one using bilinear finite elements are presented.Dedicated to Prof. Ivo Babuka on the occasion of his 60th birthdayThis research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0156 and the U.S. Army Research Office under Contract Number DAAL 03-86-K-0112     

The convergence of spline collocation for strongly elliptic equations on curves

Summary Most boundary element methods for two-dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions. Here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order. The equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines. The crucial assumption for the generalized boundary integral and integro-differential operators is strong ellipticity. Our analysis is based on simple Fourier expansions. In particular, we extend results by J. Saranen and W.L. Wendland from constant to variable coefficient equations. Our results include the first convergence proof of midpoint collocation with piecewise constant functions, i.e., the panel method for solving systems of Cauchy singular integral equations.Dedicated to Prof. Dr. Dr. h.c. mult. Lothar Collatz on the occasion of his 75th birthdayThis work was begun at the Technische Hochschule Darmstadt where Professor Arnold was supported by a North Atlantic Treaty Organization Postdoctoral Fellowship. The work of Professor Arnold is supported by NSF grant BMS-8313247. The work of Professor Wendland was supported by the Stiftung Volkswagenwerk     

Inverse Problems of Boundary Value Problems for Annular Vibrating Membranes with Piecewise Smooth Positive Functions in the Boundary Conditions

The asymptotic expansions of the trace of the heat kernel for small positive t, where λ_ν are the eigenvalues of the negative Laplacian in Rⁿ (n=2 or 3), are studied for a general annular bounded domain Ω with a smooth inner boundary ?Ω₁ and a smooth outer boundary ?Ω₂ where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the components Γ _j (j=1,…,m) of ?Ω₁ and on the components of ?Ω₂ are considered such that and and where the coefficients in the Robin boundary conditions are piecewise smooth positive functions. Some applications of Θ (t) for an ideal gas enclosed in the general annular bounded domain Ω are given.     

The numerical solution of a non-linear boundary integral equation on smooth surfaces

We study a boundary integral equation method for solving Laplace'sequation u=0 with non-linear boundary conditions. This non-linearboundary value problem is reformulated as a non-linear boundaryintegral equation, with u on the boundary as the solution beingsought. The integral equation is solved numerically by usingthe collocation method, with piecewise quadratic functions usedas approximations to u. Convergence results are given for thecases where (1) the original surface is used, and (2) the surfaceis approximated by piecewise quadratic interpolation. In addition,we define and analyze a two-grid iteration method for solvingthe non-linear system that arises from the discretization ofthe boundary integral equation. Numerical examples are given;and the paper concludes with a short discussion of the relativecost of different parts of the method. This work was supported in part by NSF grant DMS-9003287.     

Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial-boundary conditions

The Dirichlet problem is considered for a singularly perturbed parabolic reaction-diffusion equation with piecewise continuous initial-boundary conditions in a rectangular domain. The highest derivative in the equation is multiplied by a parameter ? ², ? ε (0, 1]. For small values of the parameter ?, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the limit equation passing through the point of discontinuity of the initial function, there arise a boundary layer and an interior layer (of characteristic width ?), respectively, which have bounded smoothness for fixed values of the parameter ?. Using the method of additive splitting of singularities (generated by discontinuities of the boundary function and its low-order derivatives), as well as the method of condensing grids (piecewise uniform grids condensing in a neighborhood of boundary layers), we construct and investigate special difference schemes that converge ?-uniformly with the second order of accuracy in x and the first order of accuracy in t, up to logarithmic factors.     

Korn's inequalities for piecewise vector fields

Korn's inequalities for piecewise vector fields are established. They can be applied to classical nonconforming finite element methods, mortar methods and discontinuous Galerkin methods.

     

On the lattice discrepancy of bodies of rotation with boundary points of curvature zero

This article gives an asymptotic result for the lattice point discrepancy of a large body of rotation in , whose boundary is piecewise smooth and contains points of vanishing Gaussian curvature. Received: 2 November 2006     

Mixed boundary value problems for nonlinear elliptic equations in multidimensional non-smooth domains

The nonlinear elliptic equation is investigated. It is supposed that u fulfils a mixed boundary value condition and that Ω ? IRⁿ (n ≥ 3) has a piecewise smooth boundary. W^s,2 — regularity (s < 3/2) of u and L^p — properties of the first and the second derivatives of u are proven.     

Stream vectors in three dimensional aerodynamics

Summary This work deals with the decomposition of a vector fieldu intou=×+. Non homogeneous boundary conditions on or are investigated; applications to the computation of inviscid flows are given; finally a conforming finite element implementation is studied and tested.     

Hermite interpolation by Pythagorean Hodograph space curves

We solve the problem of Hermite interpolation by Pythagorean Hodograph (PH) space curves. More precisely, for any set of space boundary data (two points with associated first and second derivatives) we construct a four-dimensional family of PH interpolants of degree and introduce a geometrically invariant parameterization of this family. This parameterization is used to identify a particular solution, which has the following properties. First, it preserves planarity, i.e., the interpolant to planar data is a planar PH curve. Second, it has the best possible approximation order 6. Third, it is symmetric in the sense that the interpolant of the ``reversed' set of boundary data is simply the ``reversed' original interpolant. This particular PH interpolant is exploited for designing algorithms for converting (possibly piecewise) analytical curves into a piecewise PH curve of degree which is globally , and for simple rational approximation of pipe surfaces with a piecewise analytical spine curve. The algorithms are presented along with an analysis of their error and approximation order.

     

Remarks on convex cones

We point out in this note that the class of cones in a locally convex topological vector space satisfying property () or piecewise relatively weakly compact cones is exactly the class of cones admitting weakly compact bases or the class of cones whose closures admit weakly compact bases.This work was supported by a Monash University Postdoctoral Fellowship.     

A new non-iterative singular sources method for the reconstruction of piecewise constant media.

Summary. We describe a novel non-iterative method for the reconstruction of a piecewise constant inhomogeneous medium in acoustic scattering, which we call the singular sources method. The basic idea of the method is to use the behaviour of the scattered field for singular incident fields (multipoles) to calculate the size of the refractive index n at some point z₀ on the boundary of the support of the scatterer and then eliminate this value from the data by subtracting a known piecewise constant background medium. The paper includes the theory for the singular sources method to locate the unknown support of an inhomogeneous medium for a known inhomogeneous background medium. Also, we give a new uniqueness proof for the reconstruction of a piecewise constant medium in two or three dimensions, using techniques that differ from those used to prove previous well-known results.Mathematics Subject Classification (2000): 35J05, 45Q05, 47A52, 78A46, 81U40Revised version received August 6, 2003