共查询到20条相似文献,搜索用时 15 毫秒
1.
Martin Petzoldt 《Advances in Computational Mathematics》2002,16(1):47-75
We consider linear elliptic equations with discontinuous coefficients in two and three space dimensions with varying boundary conditions. The problem is discretized with linear finite elements. An adaptive procedure based on a posteriori error estimators for the treatment of singularities is proposed. Within the class of quasi-monotonically distributed coefficients we derive a posteriori error estimators with bounds that are independent of the variation of the coefficients. In numerical test cases we confirm the robustness of the error estimators and observe that on adaptively refined meshes the reduction of the error is optimal with respect to the number of unknowns. 相似文献
2.
Uniform Convergence of Multigrid V-Cycle on Adaptively Refined Finite Element Meshes for Elliptic Problems with Discontinuous Coefficients 下载免费PDF全文
Haijun Wu & Weiying Zheng 《数学研究通讯:英文版》2023,39(3):437-475
The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are
considered. Under the conditions that the coefficient is quasi-monotone up to
a constant and the meshes are locally refined by using the newest vertex bisection algorithm, some uniform convergence results are proved for the standard
multigrid V-cycle algorithm with Gauss-Seidel relaxations performed only on
new nodes and their immediate neighbours. The multigrid V-cycle algorithm
uses $\mathcal{O}(N)$ operations per iteration and is optimal. 相似文献
3.
In this paper, a local multilevel algorithm is investigated for solving linear systems arising from adaptive finite element approximations of second order elliptic problems with smooth complex coefficients. It is shown that the abstract theory for local multilevel algorithm can also be applied to elliptic problems whose dominant coefficient is complex valued. Assuming that the coarsest mesh size is sufficiently small, we prove that this algorithm with Gauss-Seidel smoother is convergent and optimal on the adaptively refined meshes generated by the newest vertex bisection algorithm. Numerical experiments are reported to confirm the theoretical analysis. 相似文献
4.
Armando Coco & Giovanni Russo 《高等学校计算数学学报(英文版)》2012,5(1):19-42
In this paper we present a one dimensional second order accurate method to
solve Elliptic equations with discontinuous coefficients on an arbitrary interface. Second
order accuracy for the first derivative is obtained as well. The method is based on the
Ghost Fluid Method, making use of ghost points on which the value is defined by suitable interface conditions. The multi-domain formulation is adopted, where the problem
is split in two sub-problems and interface conditions will be enforced to close the problem. Interface conditions are relaxed together with the internal equations (following the
approach proposed in [10] in the case of smooth coefficients), leading to an iterative
method on all the set of grid values (inside points and ghost points). A multigrid approach with a suitable definition of the restriction operator is provided. The restriction
of the defect is performed separately for both sub-problems, providing a convergence
factor close to the one measured in the case of smooth coefficient and independent on
the magnitude of the jump in the coefficient. Numerical tests will confirm the second
order accuracy. Although the method is proposed in one dimension, the extension in
higher dimension is currently underway [12] and it will be carried out by combining
the discretization of [10] with the multigrid approach of [11] for Elliptic problems with
non-eliminated boundary conditions in arbitrary domain. 相似文献
5.
兴梅 《数学物理学报(A辑)》2005,25(5):685-693
该文研究的问题源自于生物学与物理学中具有间断介电系数的静电场。作者以拟微分算子为主要工具讨论具间断系数的半线性二阶椭圆型方程解的存在性和正则性 。 相似文献
6.
7.
Hongjie Dong 《分析论及其应用》2020,36(2):161-199
In this paper, we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients. We begin with an approach given by N. V. Krylov to parabolic equations in the whole space with $rm{VMO}_x$ coefficients. We then discuss some subsequent development including elliptic and parabolic equations with coefficients which are allowed to be merely measurable in one or two space directions, weighted $L_p$estimates with Muckenhoupt ($A_p$) weights, non-local elliptic and parabolic equations, as well as fully nonlinear elliptic and parabolic equations. 相似文献
8.
The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u
0 such that the linearized at u
0 problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u
0 which depends smoothly (in W
2,p
with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L
∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W
2,p
) solutions for u
0. 相似文献
9.
10.
In this paper, an extremal eigenvalue problem to the Sturm-Liouville equations with
discontinuous coefficients and volume constraint is investigated. Liouville
transformation is applied to change the problem into an equivalent minimization
problem. Finite element method is proposed and the convergence for the finite
element solution is established. A monotonic decreasing algorithm is presented
to solve the extremal eigenvalue problem. A global convergence for the algorithm
in the continuous case is proved. A few numerical results are given to depict the
efficiency of the method. 相似文献
11.
讨论了具间断系数的N维拟线性椭圆方程. 利用估计和差分逼近方法,证明了弱解的一阶导数H\"{o}lder连续到方程系数间断的内边界. 相似文献
12.
Davide Guidetti 《Mathematische Nachrichten》2002,237(1):62-88
We prove a priori estimates in Sobolev spaces for general linear elliptic boundary value problems with VMO coefficients.We give also results of existence and uniqueness of a solution. 相似文献
13.
We consider the following fractional elliptic problem: where \((-\Delta )^s, s\in (0,1)\) is the fractional Laplacian, \(\Omega \) is a bounded domain of \(\mathbb{{R}}^n,(n\ge 2s)\) with smooth boundary \(\partial \Omega ,\) H is the Heaviside step function, f is a given function and \(\mu \) is a positive real parameter. The problem (P) can be considered as simplified version of some models arising in different contexts. We employ variational techniques to study the existence and multiplicity of positive solutions of problem (P).
相似文献
$$\begin{aligned} (P)\left\{ \begin{array}{ll} (-\Delta )^s u = f(u) H(u-\mu )&{} \quad \text{ in } \ \Omega ,\\ u =0 &{}\quad \text{ on } \ \mathbb{{R}}^n {\setminus } \Omega , \end{array} \right. \end{aligned}$$
14.
We establish a global weighted W 1, p -regularity for solutions to variational inequalities and obstacle problems for divergence form elliptic systems with measurable coefficients in bounded non-smooth domains. 相似文献
15.
16.
We develop and analyze an adaptive hybridized Interior Penalty
Discontinuous Galerkin (IPDG-H) method for H(curl)-elliptic
boundary value problems in 2D or 3D arising from a
semi-discretization of the eddy currents equations. The method can
be derived from a mixed formulation of the given boundary value
problem and involves a Lagrange multiplier that is an approximation
of the tangential traces of the primal variable on the interfaces of
the underlying triangulation of the computational domain. It is
shown that the IPDG-H technique can be equivalently formulated and
thus implemented as a mortar method. The mesh adaptation is based on
a residual-type a posteriori error estimator consisting of element
and face residuals. Within a unified framework for adaptive finite
element methods, we prove the reliability of the estimator up to a
consistency error. The performance of the adaptive symmetric IPDG-H
method is documented by numerical results for representative test
examples in 2D. 相似文献
17.
Simulation of sub-surface flow in geologically complex formations is just one example in computational science, where efficient and accurate solutions of heterogeneous elliptic problems are of great interest. Often it is not feasible to resolve the whole range of relevant length scales associated with the spatial distribution of the highly varying coefficients. The MSFV method was originally developed for multi-phase flow in porous media. In the more general form presented here, it can be applied for solving large elliptic problems in various areas of computational science. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
18.
Mustafa Kandemir Oktay Mukhtarov Yakov Yakubov 《Mediterranean Journal of Mathematics》2009,6(3):317-338
In this study, a Birkhoff-irregular boundary value problem for linear ordinary differential equations of the second order
with discontinuous coefficients and the spectral parameter has been considered. Therefore, at the discontinuous point, two
additional boundary conditions (called transmission conditions) have been added to the boundary conditions. The eigenvalue
parameter is of the second degree in the differential equation and of the first degree in a boundary condition. The equation
contains an abstract linear operator which is (usually) unbounded in the space Lq(−1, 1). Isomorphism and coerciveness with defects 1 and 2 are proved for this problem. The case of the biharmonic equation
is also studied. 相似文献
19.
得到一类退化椭圆型方程弱解梯度在其拟线性系数矩阵$A(\cdot,u)$对任意$u$关于$x$一致满足VMO条件下在Morrey空间$L^{p,\lambda}$的内部正则性. 相似文献
20.
张晶 《数学物理学报(A辑)》2014,34(1):86-94
该文基于Banach空间,利用非光滑局部环绕方法考虑有界区域上具有不连续非线性项和Robin边值条件的半线性椭圆型方程解的存在性和多重性. 相似文献