共查询到20条相似文献,搜索用时 15 毫秒
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Andreas Karageorghis 《Numerical Algorithms》2009,50(2):215-240
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.
相似文献
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A. Karageorghis 《Numerical Algorithms》2013,64(3):427-453
We consider the numerical solution of certain elliptic boundary value problems in disks and annuli using the Trefftz collocation method. In particular we examine boundary value problems for the Laplace, Helmholtz, modified Helmholtz and biharmonic equations in such domains. It is shown that this approach leads to systems in which the matrices possess specific structures. By exploiting these structures we propose efficient algorithms for the solution of the systems. The proposed algorithms are applied to standard test problems. 相似文献
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Summary In this paper, methods for numerical verifications of solutions for elliptic equations in nonconvex polygonal domains are studied. In order to verify solutions using computer, it is necessary to determine some constants which appear in a priori error estimations. We propose some methods for determination of these constants. In numerical examples, calculating these constants for anL-shaped domain, we verify the solution of a nonlinear elliptic equation. 相似文献
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Boško S. Jovanovic 《Journal of Computational and Applied Mathematics》2011,236(3):364-374
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. 相似文献
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《Journal of Computational and Applied Mathematics》2012,236(3):364-374
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. 相似文献
6.
Semilinear elliptic problems in annular domains 总被引:9,自引:0,他引:9
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.
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. 相似文献
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.
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.
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. 相似文献
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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.
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Igor Kukavica 《Journal d'Analyse Mathématique》1995,67(1):269-280
We establish sharp upper bounds on the (n−1)-dimensional Hausdorff measure of the zero (nodal) sets and on the maximal order of vanishing corresponding to eigenfunctions
of a regular elliptic problem on a bounded domain Ω ⊆ ℝ
n
with real-analytic boundary. The elliptic operator may be of an arbitrary even order, and its coefficients are assumed to
be real-analytic. This extends a result of Donnelly and Fefferman ([DF1], [DF3]) concerning upper bounds for nodal volumes
of eigenfunctions corresponding to the Laplacian on compact Riemannian manifolds with boundary. 相似文献
13.
《Numerical Methods for Partial Differential Equations》2018,34(6):2316-2335
We study a new class of finite elements so‐called composite finite elements (CFEs), introduced earlier by Hackbusch and Sauter, Numer. Math., 1997; 75:447‐472, for the approximation of nonlinear parabolic equation in a nonconvex polygonal domain. A two‐scale CFE discretization is used for the space discretizations, where the coarse‐scale grid discretized the domain at an appropriate distance from the boundary and the fine‐scale grid is used to resolve the boundary. A continuous, piecewise linear CFE space is employed for the spatially semidiscrete finite element approximation and the temporal discretizations is based on modified linearized backward Euler scheme. We derive almost optimal‐order convergence in space and optimal order in time for the CFE method in the L∞(L2) norm. Numerical experiment is carried out for an L‐shaped domain to illustrate our theoretical findings. 相似文献
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Communicated by B.-W. Schulze 相似文献
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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. 相似文献
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In this paper we prove some existence results of semilinear Dirichlet problems in nonsmooth domains in presence of lower and upper solutions well-ordered or not. We first prove existence results in an abstract setting using degree theory. We secondly apply them for domains with conical points. 相似文献
19.
A. Lorenzi 《Annali dell'Universita di Ferrara》1996,41(1):85-109
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. 相似文献