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1.
Summary Finite element approximations of the eigenpairs of differential operators are computed as eigenpairs of matrices whose elements involve integrals which must be evaluated by numerical integration. The effect of this numerical integration on the eigenvalue and eigenfunction error is estimated. Specifically, for 2nd order selfadjoint eigenvalue problems we show that finite element approximations with quadrature satisfy the well-known estimates for approximations without quadrature, provided the quadrature rules have appropriate degrees of precision.The work of this author was partially supported by the National Science Foundation under Grant DMS-84-10324 相似文献
2.
For general quadrilateral or hexahedral meshes, the finite-element methods require evaluation of integrals of rational functions, instead of traditional polynomials. It remains as a challenge in mathematics to show the traditional Gauss quadratures would ensure the correct order of approximation for the numerical integration in general. However, in the case of nested refinement, the refined quadrilaterals and hexahedra converge to parallelograms and parallelepipeds, respectively. Based on this observation, the rational functions of inverse Jacobians can be approximated by the Taylor expansion with truncation. Then the Gauss quadrature of exact order can be adopted for the resulting integrals of polynomials, retaining the optimal order approximation of the finite-element methods. A theoretic justification and some numerical verification are provided in the paper. 相似文献
3.
Summary. In this paper we present a new quadrature method for computing Galerkin stiffness matrices arising from the discretisation
of 3D boundary integral equations using continuous piecewise linear boundary elements. This rule takes as points some subset
of the nodes of the mesh and can be used for computing non-singular Galerkin integrals corresponding to pairs of basis functions
with non-intersecting supports. When this new rule is combined with standard methods for the singular Galerkin integrals we
obtain a “hybrid” Galerkin method which has the same stability and asymptotic convergence properties as the true Galerkin
method but a complexity more akin to that of a collocation or Nystr?m method. The method can be applied to a wide range of
singular and weakly-singular first- and second-kind equations, including many for which the classical Nystr?m method is not
even defined. The results apply to equations on piecewise-smooth Lipschitz boundaries, and to non-quasiuniform (but shape-regular)
meshes. A by-product of the analysis is a stability theory for quadrature rules of precision 1 and 2 based on arbitrary points
in the plane. Numerical experiments demonstrate that the new method realises the performance expected from the theory.
Received January 22, 1998 / Revised version received May 26, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000 相似文献
4.
Convolution quadrature and discretized operational calculus. II 总被引:4,自引:0,他引:4
C. Lubich 《Numerische Mathematik》1988,52(4):413-425
Summary Operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations: singular integrals (power and logarithmic singularities, finite part integrals), multiple timescale convolution, Volterra integral equations, Wiener-Hopf integral equations. Frequency domain conditions, which determine, the stability of such equations, can be carried over to the discretization.This is Part II to the article with the same title (Part I), which was published in Volume 52, No. 2, pp. 129–145 (1988) 相似文献
5.
A theoretical error estimate for quadrature formulas, which depends on four approximations of the integral, is derived. We obtain a bound, often sharper than the trivial one, which requires milder conditions to be satisfied than a similar result previously presented by Laurie. A selection of numerical tests with one-dimensional integrals is reported, to show how the error estimate works in practice. 相似文献
6.
An accurate and efficient semi-analytic integration technique is developed for three-dimensional hypersingular boundary integral equations of potential theory. Investigated in the context of a Galerkin approach, surface integrals are defined as limits to the boundary and linear surface elements are employed to approximate the geometry and field variables on the boundary. In the inner integration procedure, all singular and non-singular integrals over a triangular boundary element are expressed exactly as analytic formulae over the edges of the integration triangle. In the outer integration scheme, closed-form expressions are obtained for the coincident case, wherein the divergent terms are identified explicitly and are shown to cancel with corresponding terms from the edge-adjacent case. The remaining surface integrals, containing only weak singularities, are carried out successfully by use of standard numerical cubatures. Sample problems are included to illustrate the performance and validity of the proposed algorithm. 相似文献
7.
The numerical evaluation of Hadamard finite-part integrals 总被引:2,自引:0,他引:2
D. F. Paget 《Numerische Mathematik》1981,36(4):447-453
Summary A quadrature rule is described for the numerical evaluation of Hadamard finite-part integrals with a double pole singularity within the range of integration. The rule is based upon the observation that such an integral is the derivative of a Cauchy principal value integral. 相似文献
8.
S. Nintcheu Fata 《Journal of Computational and Applied Mathematics》2011,236(6):1216-1225
A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach. 相似文献
9.
S. Nintcheu Fata 《Journal of Computational and Applied Mathematics》2011,235(15):4480-4495
On employing isoparametric, piecewise linear shape functions over a flat triangle, exact formulae are derived for all surface potentials involved in the numerical treatment of three-dimensional singular and hyper-singular boundary integral equations in linear elasticity. These formulae are valid for an arbitrary source point in space and are represented as analytical expressions along the edges of the integration triangle. They can be employed to solve integral equations defined on triangulated surfaces via a collocation method or may be utilized as analytical expressions for the inner integrals in a Galerkin technique. A numerical example involving a unit triangle and a source point located at various distances above it, as well as sample problems solved by a collocation boundary element method for the Lamé equation are included to validate the proposed formulae. 相似文献
10.
Abstract. This paper is concerned with the stability and convergence of fully discrete Galerkin methods for boundary integral equations
on bounded piecewise smooth surfaces in . Our theory covers equations with very general operators, provided the associated weak form is bounded and elliptic on , for some . In contrast to other studies on this topic, we do not assume our meshes to be quasiuniform, and therefore the analysis admits
locally refined meshes. To achieve such generality, standard inverse estimates for the quasiuniform case are replaced by appropriate
generalised estimates which hold even in the locally refined case. Since the approximation of singular integrals on or near
the diagonal of the Galerkin matrix has been well-analysed previously, this paper deals only with errors in the integration
of the nearly singular and smooth Galerkin integrals which comprise the dominant part of the matrix. Our results show how
accurate the quadrature rules must be in order that the resulting discrete Galerkin method enjoys the same stability properties
and convergence rates as the true Galerkin method. Although this study considers only continuous piecewise linear basis functions
on triangles, our approach is not restricted in principle to this case. As an example, the theory is applied here to conventional
“triangle-based” quadrature rules which are commonly used in practice. A subsequent paper [14] introduces a new and much more
efficient “node-based” approach and analyses it using the results of the present paper.
Received December 10, 1997 / Revised version received May 26, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000 相似文献
11.
In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications. 相似文献
12.
Summary We present and analyze methods for the accurate and efficient evaluation of weakly, Cauchy and hypersingular integrals over piecewise analytic curved surfaces in 3.The class of admissible integrands includes all kernels arising in the numerical solution of elliptic boundary value problems in three-dimensional domains by the boundary integral equation method. The possibly not absolutely integrable kernels of boundary integral operators in local coordinates are pseudohomogeneous with analytic characteristics depending on the local geometry of the surface at the source point. This rules out weighted quadrature approaches with a fixed singular weight.For weakly singular integrals it is shown that Duffy's triangular coordinates leadalways to a removal of the kernel singularity. Also asymptotic estimates of the integration error are provided as the size of the boundary element patch tends to zero. These are based on the Rabinowitz-Richter estimates in connection with an asymptotic estimate of domains of analyticity in 2.It is further shown that the modified extrapolation approach due to Lyness is in the weakly singular case always applicable. Corresponding error and asymptotic work estimates are presented.For the weakly singular as well as for Cauchy and hypersingular integrals which e.g. arise in the study of crack problems we analyze a family of product integration rules in local polar coordinates. These rules are hierarchically constructed from finite part integration formulas in radial and Gaussian formulas in angular direction. Again, we show how the Rabinowitz-Richter estimates can be applied providing asymptotic error estimates in terms of orders of the boundary element size.Partially supported by the Priority Research Programme Boundary Element Methods of the German Research Foundation DFG under Grant No. We 659/16-1 (guest programme) and under AFOSR-grant 89-0252. 相似文献
13.
A fictitious domain approach to the numerical solution of PDEs in stochastic domains 总被引:1,自引:0,他引:1
We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains
are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial
chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition
enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is
invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic
variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection
method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and
convergence analysis of the method, as well as numerical results, are provided. The convergence is “spectral” in the polynomial
chaos order, in any subdomain which does not contain the random boundaries. 相似文献
14.
Summary We describe a numerical method to compute free surfaces in electromagnetic shaping and levitation of liquid metals. We use an energetic variational formulation and optimization techniques to compute, a critical point. The surfaces are represented by piecewise linear finite elements. Each step of the algorithm requires solving an elliptic boundary value problem in the exterior of the intermediate surfaces. This is done by using an integral representation on these surfaces. 相似文献
15.
We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost. 相似文献
16.
17.
In this paper we use a boundary integral method with single layer potentials to solve a class of Helmholtz transmission problems
in the plane. We propose and analyze a novel and very simple quadrature method to solve numerically the equivalent system
of integral equations which provides an approximation of the solution of the original problem with linear convergence (quadratic
in some special cases). Furthermore, we also investigate a modified quadrature approximation based on the ideas of qualocation
methods. This new scheme is again extremely simple to implement and has order three in weak norms.
相似文献
18.
Saulo P. Oliveira Alexandre L. Madureira Frederic Valentin 《Journal of Computational and Applied Mathematics》2009
We discuss the numerical integration of polynomials times non-polynomial weighting functions in two dimensions arising from multiscale finite element computations. The proposed quadrature rules are significantly more accurate than standard quadratures and are better suited to existing finite element codes than formulas computed by symbolic integration. We validate this approach by introducing the new quadrature formulas into a multiscale finite element method for the two-dimensional reaction–diffusion equation. 相似文献
19.
Joseph W. Jerome 《Numerische Mathematik》2008,109(1):121-142
We consider nonlinear elliptic systems, with mixed boundary conditions, on a convex polyhedral domain Ω ⊂ R
N
. These are nonlinear divergence form generalizations of Δu = f(·, u), where f is outward pointing on the trapping region boundary. The motivation is that of applications to steady-state reaction/diffusion
systems. Also included are reaction/diffusion/convection systems which satisfy the Einstein relations, for which the Cole-Hopf
transformation is possible. For maximum generality, the theory is not tied to any specific application. We are able to demonstrate
a trapping principle for the piecewise linear Galerkin approximation, defined via a lumped integration hypothesis on integrals
involving f, by use of variational inequalities. Results of this type have previously been obtained for parabolic systems by Estep, Larson,
and Williams, and for nonlinear elliptic equations by Karátson and Korotov. Recent minimum and maximum principles have been
obtained by Jüngel and Unterreiter for nonlinear elliptic equations. We make use of special properties of the element stiffness
matrices, induced by a geometric constraint upon the simplicial decomposition. This constraint is known as the non-obtuseness
condition. It states that the inward normals, associated with an arbitrary pair of an element’s faces, determine an angle
with nonpositive cosine. Drăgănescu, Dupont, and Scott have constructed an example for which the discrete maximum principle
fails if this condition is omitted. We also assume vertex communication in each element in the form of an irreducibility hypothesis
on the off-diagonal elements of the stiffness matrix. There is a companion convergence result, which yields an existence theorem
for the solution. This entails a consistency hypothesis for interpolation on the boundary, and depends on the Tabata construction
of simple function approximation, based on barycentric regions.
This work was supported by the National Science Foundation under grant DMS-0311263. 相似文献
20.
The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition. The
existence and uniqueness of the solution of the continuous problem is established with the aid of the monotone operator theory.
The main attention is paid to the investigation of the finite element approximation using numerical integration for the computation
of nonlinear boundary integrals. The solvability of the discrete finite element problem is proved and the convergence of the
approximate solutions to the exact one is analysed.
Received April 15, 1996 / Revised version received November 22, 1996 相似文献