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1.
An important progress was recently done in numerical approximation of weak solutions to a micromagnetic model equation. The problem with the nonconvex side-constraint of preserving the length of the magnetization was tackled by using reduced integration. Several schemes were proposed and their convergence to weak solutions was proved. All schemes were derived from the Landau–Lifshitz–Gilbert form of the micromagnetic equation. However, when the precessional term in the original Landau–Lifshitz (LL) form of the micromagnetic equation tends to zero, the above schemes become unusable. 相似文献
2.
We derive residual based a posteriori error estimates of the flux in L
2-norm for a general class of mixed methods for elliptic problems. The estimate is applicable to standard mixed methods such
as the Raviart–Thomas–Nedelec and Brezzi–Douglas–Marini elements, as well as stabilized methods such as the Galerkin-Least
squares method. The element residual in the estimate employs an elementwise computable postprocessed approximation of the
displacement which gives optimal order. 相似文献
3.
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
4.
We present guaranteed and computable both sided error bounds for the discontinuous Galerkin (DG) approximations of elliptic problems. These estimates are derived in the full DG-norm on purely functional grounds by the analysis of the respective differential problem, and thus, are applicable to any qualified DG approximation. Based on the triangle inequality, the underlying approach has the following steps for a given DG approximation: (1) computing a conforming approximation in the energy space using the Oswald interpolation operator, and (2) application of the existing functional a posteriori error estimates to the conforming approximation. Various numerical examples with varying difficulty in computing the error bounds, from simple problems of polynomial-type analytic solution to problems with analytic solution having sharp peaks, or problems with jumps in the coefficients of the partial differential equation operator, are presented which confirm the efficiency and the robustness of the estimates. 相似文献
5.
An approximation scheme is defined for incompressible miscible displacement in porous media. This scheme is constructed by using two methods. Standard mixed finite element is used for the Darcy velocity equation. A characteristics-mixed finite element method is presented for the concentration equation. Characteristic approximation is applied to handle the convection part of the concentration equation, and a lowest-order mixed finite element spatial approximation is adopted to deal with the diffusion part. Thus, the scalar unknown concentration and the diffusive flux can be approximated simultaneously. In order to derive the optimal L2-norm error estimates, a post-processing step is included in the approximation to the scalar unknown concentration. This scheme conserves mass globally; in fact, on the discrete level, fluid is transported along the approximate characteristics. Numerical experiments are presented finally to validate the theoretical analysis. 相似文献
6.
Bhupen Deka 《Journal of Computational and Applied Mathematics》2010,234(2):605-612
The purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175-202]. We derive error estimates in finite element method with quadrature for elliptic interface problems in a two-dimensional convex polygonal domain. Optimal order error estimates in L2 and H1 norms are shown to hold even if the regularity of the solution is low on the whole domain. Finally, numerical experiment for two dimensional test problem is presented in support of our theoretical findings. 相似文献
7.
E. J. van Asselt 《BIT Numerical Mathematics》1985,25(2):380-385
Globally convergent nonlinear relaxation methods are considered for a class of nonlinear boundary value problems (BVPs), where the discretizations are continuousM-functions.It is shown that the equations with one variable occurring in the nonlinear relaxation methods can always be solved by Newton's method combined with the Bisection method. The nonlinear relaxation methods are used to get an initial approximation in the domain of attraction of Newton's method. Numerical examples are given. 相似文献
8.
In this paper, we propose two variants of the additive Schwarz method for the approximation of second order elliptic boundary
value problems with discontinuous coefficients, on nonmatching grids using the lowest order Crouzeix-Raviart element for the
discretization in each subdomain. The overall discretization is based on the mortar technique for coupling nonmatching grids.
The convergence behavior of the proposed methods is similar to that of their closely related methods for conforming elements.
The condition number bound for the preconditioned systems is independent of the jumps of the coefficient, and depend linearly
on the ratio between the subdomain size and the mesh size. The performance of the methods is illustrated by some numerical
results.
This work has been supported by the Alexander von Humboldt Foundation and the special funds for major state basic research
projects (973) under 2005CB321701 and the National Science Foundation (NSF) of China (No.10471144)
This work has been supported in part by the Bergen Center for Computational Science, University of Bergen 相似文献
9.
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
10.
Summary. Both mixed finite element methods and boundary integral methods are important tools in computational mechanics according to
a good stress approximation. Recently, even low order mixed methods of Raviart–Thomas-type became available for problems in
elasticity. Since either methods are robust for critical Poisson ratios, it appears natural to couple the two methods as proposed
in this paper. The symmetric coupling changes the elliptic part of the bilinear form only. Hence the convergence analysis
of mixed finite element methods is applicable to the coupled problem as well. Specifically, we couple boundary elements with
a family of mixed elements analyzed by Stenberg. The locking-free implementation is performed via Lagrange multipliers, numerical
examples are included.
Received February 21, 1995 / Revised version received December 21, 1995 相似文献
11.
Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes
Summary. We study a multilevel preconditioner for the Galerkin boundary element matrix arising from a symmetric positive-definite
bilinear form. The associated energy norm is assumed to be equivalent to a Sobolev norm of positive, possibly fractional,
order m on a bounded (open or closed) surface of dimension d, with . We consider piecewise linear approximation on triangular elements. Successive levels of the mesh are created by selectively
subdividing elements within local refinement zones. Hanging nodes may be created and the global mesh ratio can grow exponentially
with the number of levels. The coarse-grid correction consists of an exact solve, and the correction on each finer grid amounts
to a simple diagonal scaling involving only those degrees of freedom whose associated nodal basis functions overlap the refinement zone. Under appropriate assumptions on the choice of refinement zones, the condition number of the preconditioned system is shown
to be bounded by a constant independent of the number of degrees of freedom, the number of levels and the global mesh ratio.
In addition to applying to Galerkin discretisation of hypersingular boundary integral equations, the theory covers finite
element methods for positive-definite, self-adjoint elliptic problems with Dirichlet boundary conditions.
Received October 5, 2001 / Revised version received December 5, 2001 / Published online April 17, 2002
The support of this work through Visiting Fellowship grant GR/N21970 from the Engineering and Physical Sciences Research
Council of Great Britain is gratefully acknowledged. The second author was also supported by the Australian Research Council 相似文献
12.
The discretization of first kind boundary integral equations leads in general to a dense system of linear equations, whose
spectral condition number depends on the discretization used. Here we describe a general preconditioning technique based on
a boundary integral operator of opposite order. The corresponding spectral equivalence inequalities are independent of the
special discretization used, i.e., independent of the triangulations and of the trial functions. Since the proposed preconditioning
form involves a (pseudo)inverse operator, one needs for its discretization only a stability condition for obtaining a spectrally
equivalent approximation.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
13.
In this paper we propose a hybrid between direct and indirect boundary integral methods to solve a transmission problem for the Helmholtz equation in Lipschitz and smooth domains. We present an exhaustive abstract study of the numerical approximation of the resulting system of boundary integral equations by means of Galerkin methods. Some particular examples of convergent schemes in the smooth case in two dimensions are given. Finally, we extend the results to a thermal scattering problem in a half plane with several obstacles and provide numerical results that illustrate the accuracy of our methods depending on the regularity of the interface. 相似文献
14.
Barbara I. Wohlmuth 《Numerische Mathematik》1999,84(1):143-171
Summary. A residual based error estimator for the approximation of linear elliptic boundary value problems by nonconforming finite
element methods is introduced and analyzed. In particular, we consider mortar finite element techniques restricting ourselves
to geometrically conforming domain decomposition methods using P1 approximations in each subdomain. Additionally, a residual
based error estimator for Crouzeix-Raviart elements of lowest order is presented and compared with the error estimator obtained
in the more general mortar situation. It is shown that the computational effort of the error estimator can be considerably
reduced if the special structure of the Lagrange multiplier is taken into account.
Received July 18, 1997 / Revised version received July 27, 1998 / Published online September 7, 1999 相似文献
15.
Composite finite elements for the approximation of PDEs on domains with complicated micro-structures
Summary. Usually, the minimal dimension of a finite element space is closely related to the geometry of the physical object of interest.
This means that sometimes the resolution of small micro-structures in the domain requires an inadequately fine finite element
grid from the viewpoint of the desired accuracy. This fact limits also the application of multi-grid methods to practical
situations because the condition that the coarsest grid should resolve the physical object often leads to a huge number of
unknowns on the coarsest level. We present here a strategy for coarsening finite element spaces independently of the shape
of the object. This technique can be used to resolve complicated domains with only few degrees of freedom and to apply multi-grid
methods efficiently to PDEs on domains with complex boundary. In this paper we will prove the approximation property of these
generalized FE spaces.
Received June 9, 1995 / Revised version received February 5, 1996 相似文献
16.
A modification of the multigrid method for the solution of linear algebraic equation systems with a strongly nonsymmetric matrix obtained after difference approximation of the convection-diffusion equation with dominant convection is proposed. Specially created triangular iterative methods have been used as the smoothers of the multigrid method. Some theoretical and numerical results are presented. 相似文献
17.
Summary In this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior Dirichlet problem in the plane. Specifically, the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region, and the Laplace equation in the corresponding unbounded exterior region, in addition to appropriate boundary and transmission conditions. The main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation. We provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence, uniqueness and approximation results are established. Then, we consider the case where the corresponding operator is strongly monotone and Lipschitz-continuous, and derive asymptotic error estimates for a boundary-finite element solution. We prove the unique solvability of the discrete operator equations, and based on a Strang type abstract error estimate, we show the strong convergence of the approximated solutions. Moreover, under additional regularity assumptions on the solution of the continous operator equation, the asymptotic rate of convergenceO (h) is obtained.The first author's research was partly supported by the U.S. Army Research Office through the Mathematical Science Institute of Cornell University, by the Universidad de Concepción through the Facultad de Ciencias, Dirección de Investigación and Vicerretoria, and by FONDECYT-Chile through Project 91-386. 相似文献
18.
We discuss a choice of weight in penalization methods. The motivation for the use of penalization in computational mathematics
is to improve the conditioning of the numerical solution. One example of such improvement is a regularization, where a penalization
substitutes an ill-posed problem for a well-posed one. In modern numerical methods for PDEs a penalization is used, for example,
to enforce a continuity of an approximate solution on non-matching grids. A choice of penalty weight should provide a balance
between error components related with convergence and stability, which are usually unknown. In this paper we propose and analyze
a simple adaptive strategy for the choice of penalty weight which does not rely on a priori estimates of above mentioned components.
It is shown that under natural assumptions the accuracy provided by our adaptive strategy is worse only by a constant factor
than one could achieve in the case of known stability and convergence rates. Finally, we successfully apply our strategy for
self-regularization of Volterra-type severely ill-posed problems, such as the sideways heat equation, and for the choice of
a weight in interior penalty discontinuous approximation on non-matching grids. Numerical experiments on a series of model
problems support theoretical results. 相似文献
19.
Summary. In this paper we consider two aspects of the problem of designing efficient numerical methods for the approximation of semilinear
boundary value problems. First we consider the use of two and multilevel algorithms for approximating the discrete solution.
Secondly we consider adaptive mesh refinement based on feedback information from coarse level approximations. The algorithms
are based on an a posteriori error estimate, where the error is estimated in terms of computable quantities only. The a posteriori
error estimate is used for choosing appropriate spaces in the multilevel algorithms, mesh refinements, as a stopping criterion
and finally it gives an estimate of the total error.
Received April 8, 1997 / Revised version received July 27, 1998 / Published online September 24, 1999 相似文献
20.
This paper deals with the numerical simulation of the steady state two dimensional window Josephson junctions by finite element method. The model is represented by a sine-Gordon type composite PDE problem. Convergence and error analysis of the finite element approximation for this semilinear problem are presented. An efficient and reliable Newton-preconditioned conjugate gradient algorithm is proposed to solve the resulting nonlinear discrete system. Regular solution branches are computed using a simple continuation scheme. Numerical results associated with interesting physical phenomena are reported. Interface relaxation methods, which by taking advantage of special properties of the composite PDE, can further reduce the overall computational cost are proposed. The implementation and the associated numerical experiments of a particular interface relaxation scheme are also presented and discussed. 相似文献