The planar radiosity equation and its numerical solution

This article gives properties of the planar radiosity equationand methods for its numerical solution. Regularity propertiesof the radiosity solution are examined, including both the effectsof corners and the effects of the visibility function. Theseare taken into account in the design of collocation methodswith piecewise polynomial approximating functions. Numericalexamples conclude the paper.     

Edge asymptotics for the radiosity equation over polyhedral boundaries

In the present paper we consider the radiosity equation over the boundary of a polyhedral domain. Similarly to corresponding results on the double‐layer potential equation, the solution of the second kind integral equation with non‐compact integral operator is piecewise continuous. The partial derivatives, however, are not bounded. In the present paper we derive the first term in the asymptotic expansion of the solution in the vicinity of an edge. Note that, knowing this term, optimal mesh gradings can be designed for the numerical solution of this equation. Copyright © 1999 John Wiley & Sons, Ltd.     

Comparing variational methods for the hinged Kirchhoff plate with corners

The hinged Kirchhoff plate model contains a fourth order elliptic differential equation complemented with a zeroeth and a second order boundary condition. On domains with boundaries having corners the strong setting is not well‐defined. We here allow boundaries consisting of piecewise C^{2, 1}‐curves connecting at corners. For such domains different variational settings will be discussed and compared. As was observed in the so‐called Saponzhyan–Babushka paradox, domains with reentrant corners need special care. In that case, a variational setting that corresponds to a second order system needs an augmented solution space in order to find a solution in the appropriate Sobolev‐type space.     

A fast collocation method for the radiosity equation,based on the hierarchical algorithm of Hanrahan and Salzman: the 1d case

In this paper we study a collocation method with piecewise constant trial functions for the solution of the planar radiosity equation. The matrix of the collocation method is approximated by a method which was developed by Hanrahan et al. [9,10]. We prove that the modified collocation method results in a reduction of work while the order of convergence stays the same. Numerical examples demonstrate the theoretical results. AMS subject classification 45B05, 65R20, 65Y20     

Extrapolation for the Approximations to the Solution of a Boundary Integral Equation on Polygonal Domains

In this paper, we consider a boundary integral equation of second kind rising from potential theory. The equation may be solved numerically by Galerkin's method using piecewise constant functions. Because of the singularities produced by the corners, we have to grade the mesh near the corner. In general, Chandler obtained the order 2 superconvergence of the iterated Galerkin solution in the uniform norm. It is proved in this paper that the Richardson extrapolation increases the accuracy from order 2 to order 4.     

Parameter-uniform error estimate for the implicit four-point scheme for a singularly perturbed heat equation with corner singularities

We study an initial-boundary value problem for a singularly perturbed one-dimensional heat equation on an interval. At the corner points, the input data are subjected to continuity conditions only, which violates the smoothness of the derivatives of the solution in neighborhoods of these points, starting from the derivatives occurring in the equation. To approximate the problem, we use the implicit four-point difference scheme on a Shishkin grid uniform with respect to time and piecewise uniform with respect to the space variable. We prove that the grid solution error is O(τ +N ^?2 ln² N) ln(j +1) uniformly with respect to the parameter, where τ is the grid increment with respect to the time variable, j is the index of the time layer, and N is the number of nodes in the piecewise uniform space grid.     

When are the spatial level surfaces of solutions of diffusion equations invariant with respect to the time variable?

We consider solutions of the initial-Neumann problem for the heat equation on bounded Lipschitz domains in ℝ ^N and classify the solutions whose spatial level surfaces are invariant with respect to the time variable. (Of course, the values of each solution on its spatial level surfaces vary with time.) The prototype of such classification is a result of Alessandrini, which proved a conjecture of Klamkin. He considered the initial-Dirichlet problem for the heat equation on bounded domains and showed that if all the spatial level surfaces of the solution are invariant with respect to the time variable under the homogeneous Dirichlet boundary condition, then either the initial data is an eigenfunction or the domain is a ball and the solution is radially symmetric with respect to the space variable. His proof is restricted to the initial-Dirichlet problem for the heat equation. In the present paper, in order to deal with the initial-Neumann problem, we overcome this obstruction by using the invariance condition of spatial level surfaces more intensively with the help of the classification theorem ofisoparametric hypersurfaces in Euclidean space of Levi-Civita and Segre. Furthermore, we can deal with nonlinear diffusion equations, such as the porous medium equation.     

Asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries

In this paper, we investigate the asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries. Under some suitable assumptions, we prove that the solution approaches a combination of Lipschitz continuous and piecewise C¹ traveling wave solution. As an application, we apply the result to the equation for time-like extremal surfaces in the Minkowski space-time R1+(1+n).     

Conform and mixed finite element schemes for the Dirichlet problem for the Bilaplacian in plane domains with corners

In plane domains with corners for the Bilaplacian a uniquely solvable conform variational principle is studied on weighted Sobolev spaces which is equivalent to the standard Dirichlet problem in the weak form. Clamped plates under point forces near corners are handled by this approach. With weighted Hsieh-Clough-Tocher elements on regular triangulations as conform C¹-finite elements a new error analysis is performed without higher regularity assumptions on the exact solution than given by the data and the boundary. The rate of convergence of the error depends on the eigenvalue with smallest imaginary part of a clamped infinite wedge since this eigenvalue describes the singularity of the exact solution in a sector with same angle. Using different spaces of trial and test functions in the standard Galerkin procedure it is shown that the error in the weighted energy norm does not pollute. For convex corners asymptotic error estimates, are proved yielding convergence for a mixed method in hydrodynamics where the solution of a system of 2^nd order and its Laplacian are approximated simultaneously by C⁰-finite elements being piecewise polynomials.     

On a quadrature algorithm for the piecewise linear wavelet collocation applied to boundary integral equations

In this paper, we consider a piecewise linear collocation method for the solution of a pseudo‐differential equation of order r=0, ?1 over a closed and smooth boundary manifold. The trial space is the space of all continuous and piecewise linear functions defined over a uniform triangular grid and the collocation points are the grid points. For the wavelet basis in the trial space we choose the three‐point hierarchical basis together with a slight modification near the boundary points of the global patches of parametrization. We choose linear combinations of Dirac delta functionals as wavelet basis in the space of test functionals. For the corresponding wavelet algorithm, we show that the parametrization can be approximated by low‐order piecewise polynomial interpolation and that the integrals in the stiffness matrix can be computed by quadrature, where the quadrature rules are composite rules of simple low‐order quadratures. The whole algorithm for the assembling of the matrix requires no more than O(N [logN]³) arithmetic operations, and the error of the collocation approximation, including the compression, the approximative parametrization, and the quadratures, is less than O(N^?(2?r)/2). Note that, in contrast to well‐known algorithms by Petersdorff, Schwab, and Schneider, only a finite degree of smoothness is required. In contrast to an algorithm of Ehrich and Rathsfeld, no multiplicative splitting of the kernel function is required. Beside the usual mapping properties of the integral operator in low order Sobolev spaces, estimates of Calderón–Zygmund type are the only assumptions on the kernel function. Copyright © 2003 John Wiley & Sons, Ltd.     

On the stability of the Collocation method for the Radiosity equation on polyhedral domains

This paper investigates the stability of the collocation methodfor the radiosity equation. We introduce graded meshes and trialspaces of piecewise polynomials and prove stability for a modifiedcollocation method. Graded meshes are necessary for higher convergencerates. The generation of triangulations which allow higher-orderapproximations leads to geometrical problems which are interestingin themselves, but do not affect the stability of the collocationmethod.     

An affine interpretation of Bäcklund transformations

The paper is devoted to an affine interpretation of Bäcklundmaps (Bäcklund transformations are a particular case of Bäcklund maps) for second order differential equations with unknown function of two arguments. Note that up to now there are no papers where Bäcklund transformations are interpreted as transformations of surfaces in a space other than Euclidean space. In this paper, we restrict our considerations to the case of so-called Bäcklund maps of class 1. The solutions of a differential equation are represented as surfaces of an affine space with induced connection determining a representation of zero curvature. We show that, in the case when a second order partial differential equation admits a Bäcklund map of class 1, for each solution of the equation there is a congruence of straight lines in an affine space formed by the tangents to the affine image of the solution. This congruence is an affine analog of a parabolic congruence in Euclidean space. The Bäcklund map can be interpreted as a transformation of surfaces of an affine space under which the affine image of a solution of the differential equation is mapped into a particular boundary surface of the congruence.     

The Numerical Solution of Laplace's Equation on a Wedge

Laplace's equation is considered on regions in the plane, withthe boundary having corners; and the double-layer potentialis used to derive a solution. The essential difficulties, boththeoretically and numerically, are reduced to the case in whichthe boundary is a simple open wedge. The theoretical behaviourof the double layer integral equation is studied explicitly,and then piecewise linear and piecewise quadratic collocationmethods are applied to the numerical solution of the equation.The major question of interest is the stability of the inversesof the approximating equations. The behaviour of the numericalmethods is somewhat surprising, and it is much better than pastanalyses would have led one to expect.     

Parabolic finite element equations in nonconvex polygonal domains

Let Ω be a bounded nonconvex polygonal domain in the plane. Consider the initial boundary value problem for the heat equation with homogeneous Dirichlet boundary conditions and semidiscrete and fully discrete approximations of its solution by piecewise linear finite elements in space. The purpose of this paper is to show that known results for the stationary, elliptic, case may be carried over to the time dependent parabolic case. A special feature in a polygonal domain is the presence of singularities in the solutions generated by the corners even when the forcing term is smooth. These cause a reduction of the convergence rate in the finite element method unless refinements are employed.     

Existence and regularity of the solution of a mixed boundary value problem for the Keldysh equation with a nonlinear absorption term

The Keldysh equation is a more general form of the classic Tricomi equation from fluid dynamics. Its well-posedness and the regularity of its solution are interesting and important. The Keldysh equation is elliptic in y>0 and is degenerate at the line y=0 in R². Adding a special nonlinear absorption term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain—similar to the potential fluid shock reflection problem. By means of an elliptic regularization technique, a delicate a priori estimate and compact argument, we show that the solution of a mixed boundary value problem of the Keldysh equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary under some conditions. We believe that this kind of regularity result for the solution will be rather useful.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> error estimate of the finite volume element methods on quadrilateral meshes

In this paper the optimal L ² error estimates of the finite volume element methods (FVEM) for Poisson equation are discussed on quadrilateral meshes. The trial function space is taken as isoparametric bilinear finite element space on quadrilateral partition, and the test function space is defined as piecewise constant space on dual partition. Under the assumption that all elements on quadrilateral meshes are O(h ²) quasi-parallel quadrilateral elements, we prove convergence rate to be O(h ²) in L ² norm.     

On the Spectrum of the Reflection Operator on Conical Surfaces

We study the mapping properties of the reflection operator on a conical surface. This allows us to derive regularity results for the solution of the radiosity equation on conical surfaces in a scale of weighted Sobolev spaces. To motivate the calculations we first study the operator on a cylinder. Here we estimate the asymptotic behavior of the spectrum of the reflection operator by partial integration. This method works also for the conical case, but first we have to find a simple representation for some hypergeometric functions.     

Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility

We consider the Cahn-Hilliard equation with a logarithmic free energy and non-degenerate concentration dependent mobility. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions. Finally some numerical experiments are presented.

     

The Policy Iteration Algorithm for Average Continuous Control of Piecewise Deterministic Markov Processes

The main goal of this paper is to apply the so-called policy iteration algorithm (PIA) for the long run average continuous control problem of piecewise deterministic Markov processes (PDMP’s) taking values in a general Borel space and with compact action space depending on the state variable. In order to do that we first derive some important properties for a pseudo-Poisson equation associated to the problem. In the sequence it is shown that the convergence of the PIA to a solution satisfying the optimality equation holds under some classical hypotheses and that this optimal solution yields to an optimal control strategy for the average control problem for the continuous-time PDMP in a feedback form.     

The Invertibility of the Double Layer Potential Operator in the Space of Continuous Functions Defined on a Polyhedron: The Panel Method

We consider the double layer potential operator W defined on the polyhedral boundary of an infinite cone and prove the invertibility of (I±2W) in the space of continuous functions. To do this we define an operator-valued symbol function for W and show that the spectral radii of its values are less than one half. In the last part of this paper we consider a piecewise constant collocation method for the numerical solution of the double layer potential equation over the boundary of a bounded polyhedron.