Kansa-RBF algorithms for elliptic problems in regular polygonal domains

We propose matrix decomposition algorithms for the efficient solution of the linear systems arising from Kansa radial basis function discretizations of elliptic boundary value problems in regular polygonal domains. These algorithms exploit the symmetry of the domains of the problems under consideration which lead to coefficient matrices possessing block circulant structures. In particular, we consider the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. Numerical examples demonstrating the applicability of the proposed algorithms are presented.     

Convergence of a FEM and two-grid algorithms for elliptic problems on disjoint domains

In this paper, we analyze a FEM and two-grid FEM decoupling algorithms for elliptic problems on disjoint domains. First, we study the rate of convergence of the FEM and, in particular, we obtain a superconvergence result. Then with proposed algorithms, the solution of the multi-component domain problem (simple example — two disjoint rectangles) on a fine grid is reduced to the solution of the original problem on a much coarser grid together with solution of several problems (each on a single-component domain) on fine meshes. The advantage is the computational cost although the resulting solution still achieves asymptotically optimal accuracy. Numerical experiments demonstrate the efficiency of the algorithms.     

A sixth-order optimal collocation method for elliptic problems

In this paper, we present a collocation method based on biquintic splines for a fourth order elliptic problems. To have a better accuracy, we formulate the standard collocation method by an appropriate perturbation on the original differential equations that leads to an optimal approximating scheme. As a result, computational results confirm that this method is optimal.     

Convergence of a FEM and two-grid algorithms for elliptic problems on disjoint domains

In this paper, we analyze a FEM and two-grid FEM decoupling algorithms for elliptic problems on disjoint domains. First, we study the rate of convergence of the FEM and, in particular, we obtain a superconvergence result. Then with proposed algorithms, the solution of the multi-component domain problem (simple example — two disjoint rectangles) on a fine grid is reduced to the solution of the original problem on a much coarser grid together with solution of several problems (each on a single-component domain) on fine meshes. The advantage is the computational cost although the resulting solution still achieves asymptotically optimal accuracy. Numerical experiments demonstrate the efficiency of the algorithms.     

A highly accurate collocation Trefftz method for solving the Laplace equation in the doubly connected domains

A highly accurate new solver is developed to deal with the Dirichlet problems for the 2D Laplace equation in the doubly connected domains. We introduce two circular artificial boundaries determined uniquely by the physical problem domain, and derive a Dirichlet to Dirichlet mapping on these two circles, which are exact boundary conditions described by the first kind Fredholm integral equations. As a direct result, we obtain a modified Trefftz method equipped with two characteristic length factors, ensuring that the new solver is stable because the condition number can be greatly reduced. Then, the collocation method is used to derive a linear equations system to determine the unknown coefficients. The new method possesses several advantages: mesh‐free, singularity‐free, non‐illposedness, semi‐analyticity of solution, efficiency, accuracy, and stability. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Semilinear elliptic problems in annular domains

The method of shooting is used to establish existence of positive radially symmetric solutions to nonlinear elliptic equations of the form u+f(r, u)=0 on annular regionsa<r=|x|<b inR ^N , satisfying Dirichlet or Neumann conditions on the boundary. This extends recent work done by Bandle, Coffman and Marcus. A result concerning uniqueness of such solutions is also extended.

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Zusammenfassung Mit Hilfe eines Schiessverfahrens wird die Existenz von Lösungen nichtlinearer Probleme der Form u+f(r, u)=0 in ringförmigen Gebieten nachgewiesen, die verschiedenen Randbedingungen genügen. Es wird auch ihre Eindeutigkeit untersucht. Diese Arbeit verallgemeinert gewisse Ergebnisse von Bandle, Coffman und Marcus.

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Résumé On utilise une méthode de tir pour établir l'existence de solutions de problèmes non linéaires du type u+f(r, u)=0 dans des anneaux, vérifiant différentes conditions aux limites. Ensuite on discute l'unicité de ces solutions. Ce travail généralise certains résultats de Bandle, Coffman et Marcus.

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This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38. The author also wants to thank the Mathematics Institute, University of Basel, for supporting a visit in Oct., 1987 during which the project was initiated.     

Adaptive wavelet algorithms for elliptic PDE's on product domains

With standard isotropic approximation by (piecewise) polynomials of fixed order in a domain , the convergence rate in terms of the number of degrees of freedom is inversely proportional to the space dimension . This so-called curse of dimensionality can be circumvented by applying sparse tensor product approximation, when certain high order mixed derivatives of the approximated function happen to be bounded in . It was shown by Nitsche (2006) that this regularity constraint can be dramatically reduced by considering best -term approximation from tensor product wavelet bases. When the function is the solution of some well-posed operator equation, dimension independent approximation rates can be practically realized in linear complexity by adaptive wavelet algorithms, assuming that the infinite stiffness matrix of the operator with respect to such a basis is highly compressible. Applying piecewise smooth wavelets, we verify this compressibility for general, non-separable elliptic PDEs in tensor domains. Applications of the general theory developed include adaptive Galerkin discretizations of multiple scale homogenization problems and of anisotropic equations which are robust, i.e., independent of the scale parameters, resp. of the size of the anisotropy.

     

Efficient algorithms for orthogonal packing problems

The NP complete problem of the orthogonal packing of objects of arbitrary dimension is considered in the general form. A new model for representing objects in containers is proposed that ensures the fast design of an orthogonal packing. New heuristics for the placement of orthogonal packing are proposed. A single-pass heuristic algorithm and a multimethod genetic algorithm are developed that optimize an orthogonal packing solution by increasing the packing density. Numerical experiments for two- and three-dimensional orthogonal packing problems are performed.     

Efficient Kansa-type MFS algorithm for elliptic problems

In this study we propose an efficient Kansa-type method of fundamental solutions (MFS-K) for the numerical solution of certain problems in circular geometries. In particular, we consider problems governed by the inhomogeneous Helmholtz equation in disks and annuli. The coefficient matrices in the linear systems resulting from the MFS-K discretization of these problems possess a block circulant structure and can thus be solved by means of a matrix decomposition algorithm and fast Fourier Transforms. Several numerical examples demonstrating the efficacy of the proposed algorithm are presented.     

Wavelets collocation methods for the numerical solution of elliptic BV problems

Based on collocation with Haar and Legendre wavelets, two efficient and new numerical methods are being proposed for the numerical solution of elliptic partial differential equations having oscillatory and non-oscillatory behavior. The present methods are developed in two stages. In the initial stage, they are developed for Haar wavelets. In order to obtain higher accuracy, Haar wavelets are replaced by Legendre wavelets at the second stage. A comparative analysis of the performance of Haar wavelets collocation method and Legendre wavelets collocation method is carried out. In addition to this, comparative studies of performance of Legendre wavelets collocation method and quadratic spline collocation method, and meshless methods and Sinc–Galerkin method are also done. The analysis indicates that there is a higher accuracy obtained by Legendre wavelets decomposition, which is in the form of a multi-resolution analysis of the function. The solution is first found on the coarse grid points, and then it is refined by obtaining higher accuracy with help of increasing the level of wavelets. The accurate implementation of the classical numerical methods on Neumann’s boundary conditions has been found to involve some difficulty. It has been shown here that the present methods can be easily implemented on Neumann’s boundary conditions and the results obtained are accurate; the present methods, thus, have a clear advantage over the classical numerical methods. A distinct feature of the proposed methods is their simple applicability for a variety of boundary conditions. Numerical order of convergence of the proposed methods is calculated. The results of numerical tests show better accuracy of the proposed method based on Legendre wavelets for a variety of benchmark problems.     

A collocation method for parabolic quasilinear problems on general domains

A collocation method is described which obtains approximate solutions to quasilinear parabolic problems on a general two-dimensional domain. The method is best suited for obtaining robust solutions to smooth problems with the accuracy required in most engineering applications. The solution is obtained in terms of a finite element, B-spline basis. An interactive computer graphics system is used for both problem formulation and the subsequent display of selected results. The theoretical basis for the method is discussed, and some typical computational results are presented.     

Efficient MFS algorithms in regular polygonal domains

In this work, we apply the Method of Fundamental Solutions (MFS) to harmonic and biharmonic problems in regular polygonal domains. The matrices resulting from the MFS discretization possess a block circulant structure. This structure is exploited to produce efficient Fast Fourier Transform–based Matrix Decomposition Algorithms for the solution of these problems. The proposed algorithms are tested numerically on several examples.      

Sparse second moment analysis for elliptic problems in stochastic domains

We consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations. Assuming normal perturbations with small amplitude and known mean field and two-point correlation function, we derive, using a second order shape calculus, deterministic equations for the mean field and the two-point correlation function of the random solution for a model Dirichlet problem which are 3rd order accurate in the boundary perturbation size. Using a variational boundary integral equation formulation on the unperturbed, “nominal” boundary and a wavelet discretization, we present and analyze an algorithm to approximate the random solution’s mean and its two-point correlation function at essentially optimal order in essentially work and memory, where N denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain. This work was supported by the EEC Human Potential Programme under contract HPRN-CT-2002-00286, “Breaking Complexity.” Work initiated while HH visited the Seminar for Applied Mathematics at ETH Zürich in the Wintersemester 2005/06 and completed during the summer programme CEMRACS2006 “Modélisation de l’aléatoire et propagation d’incertitudes” in July and August 2006 at the C.I.R.M., Marseille, France.     

A factorization method for elliptic problems in a circular domain

We present a method to factorize a second order elliptic boundary value problem in a circular domain, in a system of uncoupled first order initial value problems. We use a space invariant embedding technique along the radius of the circle, in a decreasing way. This technique is inspired in the temporal invariant embedding used by J.-L. Lions for the control of parabolic systems. The singularity at the origin for the initial value problems is studied. A formal calculation for more general star-shaped domains is presented. To cite this article: J. Henry et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Direct and inverse problems for elliptic equations in cylindrical domains

We consider direct and inverse boundary value problems for elliptic equations in divergence form related to cylindrical domains with a smooth lateral surface. Our basic assumptions is that the differential operator may be represented as a sum of two differential operators in divergence form, the former acting on the «transversal» variables only, the latter on the «axial» one only. Slightly extending well-known abstract results in [4], we can prove an existence-uniqueness and continuous dependence result for the direct problem. This allows to show an existence theorem for the inverse problem, when the additional unknown is a «conductivity» coefficient depending on the axial variable, only.