共查询到20条相似文献，搜索用时 15 毫秒
1.
Gabriel N. Gatica Matthias Maischak 《Numerical Methods for Partial Differential Equations》2005,21(3):421-450
We consider the mixed finite element method with Lagrange multipliers as applied to second‐order elliptic equations in divergence form with mixed boundary conditions. The corresponding Galerkin scheme is defined by using Raviart‐Thomas spaces. We develop a posteriori error analyses yielding a reliable and efficient estimate based on residuals, and a reliable and quasi‐efficient estimate based on local problems, respectively. Several numerical results illustrate the suitability of these a posteriori estimates for the adaptive computation of the discrete solutions. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 相似文献
2.
Min Yang 《Numerical Methods for Partial Differential Equations》2011,27(2):277-291
In this article, we study the a posteriori H1 and L2 error estimates for Crouzeix‐Raviart nonconforming finite volume element discretization of general second‐order elliptic problems in ?2. The error estimators yield global upper and local lower bounds. Finally, numerical experiments are performed to illustrate the theoretical findings. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011 相似文献
3.
Zhangxin Chen 《Numerical Methods for Partial Differential Equations》1997,13(5):483-503
The recently proposed expanded mixed formulation for numerical solution of second-order elliptic problems is here extended to fourth-order elliptic problems. This expanded formulation for the differential problems under consideration differs from the classical formulation in that three variables are treated, i.e., the displacement, the stress, and the moment tensors. It works for the case where the coefficient of the differential equations is small and does not need to be inverted, or for the case in which the stress tensor of the equations does not need to be symmetric. Based on this new formulation, various mixed finite elements for fourth-order problems are considered; error estimates of quasi-optimal or optimal order depending upon the mixed elements are derived. Implementation techniques for solving the linear system arising from these expanded mixed methods are discussed, and numerical results are presented. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 483–503, 1997 相似文献
4.
A least‐squares mixed finite element (LSMFE) schemes are formulated to solve the 1D regularized long wave (RLW) equations and the convergence is discussed. The L2 error estimates of LSMFE methods for RLW equations under the standard regularity assumption on the finite element partition are given.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 相似文献
5.
We treat the finite volume element method (FVE) for solving general second order elliptic problems as a perturbation of the linear finite element method (FEM), and obtain the optimal H1 error estimate, H1 superconvergence and Lp (1 < p ≤ ∞) error estimates between the solution of the FVE and that of the FEM. In particular, the superconvergence result does not require any extra assumptions on the mesh except quasi‐uniform. Thus the error estimates of the FVE can be derived by the standard error estimates of the FEM. Moreover we consider the effects of numerical integration and prove that the use of barycenter quadrature rule does not decrease the convergence orders of the FVE. The results of this article reveal that the FVE is in close relationship with the FEM. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 693–708, 2003. 相似文献
6.
Ivo Babu&#x;ka Gabriel N. Gatica 《Numerical Methods for Partial Differential Equations》2003,19(2):192-210
In this note we analyze a modified mixed finite element method for second‐order elliptic equations in divergence form. As a model we consider the Poisson problem with mixed boundary conditions in a polygonal domain of R 2. The Neumann (essential) condition is imposed here in a weak sense, which yields the introduction of a Lagrange multiplier given by the trace of the solution on the corresponding boundary. This approach allows to handle nonhomogeneous Neumann boundary conditions, theoretically and computationally, in an alternative and usually easier way. Then we utilize the classical Babu?ka‐Brezzi theory to show that the resulting mixed variational formulation is well posed. In addition, we use Raviart‐Thomas spaces to define the associated finite element method and, applying some elliptic regularity results, we prove the stability, unique solvability, and convergence of this discrete scheme, under appropriate assumptions on the mesh sizes. Finally, we provide numerical results illustrating the performance of the algorithm for smooth and singular problems. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 192–210, 2003 相似文献
7.
The expanded mixed covolume method for the two‐dimensional Sobolev equation with convection term is developed and studied. This method uses the lowest‐order Raviart‐Thomas mixed finite element space as the trial function space. By introducing a transfer operator γh which maps the trial function space into the test function space and combining expanded mixed finite element with mixed covolume method, the continuous‐in‐time, discrete‐in‐time expanded mixed covolume schemes are constructed, and optimal error estimates for these schemes are obtained. Numerical results are given to examine the validity and effectiveness of the proposed schemes.© 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
8.
In this paper, we will investigate the error estimates and thesuperconvergence property of mixed finite element methods for asemilinear elliptic control problem with an integral constraint oncontrol. The state and co-state are approximated by the lowest orderRaviart-Thomas mixed finite element and the control variableis approximated by piecewise constant functions. We derive somesuperconvergence properties for the control variable and the statevariables. Moreover, we derive $L^∞$- and $H^{-1}$-errorestimates both for the control variable and the state variables.Finally, a numerical example is given to demonstrate the theoreticalresults. 相似文献
9.
Kwang Y. Kim 《Numerical Methods for Partial Differential Equations》2005,21(4):791-809
In this article we construct and analyze a mixed finite volume method for second‐order nonlinear elliptic problems employing H(div; Ω)‐conforming approximations for the vector variable and completely discontinuous approximations for the scalar variable. The main attractive feature of our method is that, although the vector variable is H(div; Ω)‐conforming, one can eliminate it in a local manner to obtain a discontinuous Galerkin method for the scalar variable. Optimal error estimates will be established for both vector and scalar variables. We also present a fully discrete version of this method that is more convenient for computational purposes. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 相似文献
10.
Sarvesh Kumar 《Numerical Methods for Partial Differential Equations》2012,28(4):1354-1381
The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations, the pressure‐velocity equation and the concentration equation. In this article, we present a mixed finite volume element method for the approximation of pressure‐velocity equation and a discontinuous Galerkin finite volume element method for the concentration equation. A priori error estimates in L∞(L2) are derived for velocity, pressure, and concentration. Numerical results are presented to substantiate the validity of the theoretical results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012 相似文献
11.
Ilya D. Mishev Qian‐Yong Chen 《Numerical Methods for Partial Differential Equations》2007,23(5):1122-1138
We derive a novel finite volume method for the elliptic equation, using the framework of mixed finite element methods to discretize the pressure and velocities on two different grids (covolumes), triangular (tetrahedral) mesh and control volume mesh. The new discretization is defined for tensor diffusion coefficient and well suited for heterogeneous media. When the control volumes are created by connecting the center of gravity of each triangle to the midpoints of its edges, we show that the discretization is stable and first order accurate for both scalar and vector unknowns. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
12.
A general mixed covolume framework for constructing conservative schemes for elliptic problems 总被引：2，自引：0，他引：2
We present a general framework for the finite volume or covolume schemes developed for second order elliptic problems in mixed form, i.e., written as first order systems. We connect these schemes to standard mixed finite element methods via a one-to-one transfer operator between trial and test spaces. In the nonsymmetric case (convection-diffusion equation) we show one-half order convergence rate for the flux variable which is approximated either by the lowest order Raviart-Thomas space or by its image in the space of discontinuous piecewise constants. In the symmetric case (diffusion equation) a first order convergence rate is obtained for both the state variable (e.g., concentration) and its flux. Numerical experiments are included.
13.
研究了参数识别问题混合有限元解的最大模误差估计.利用1阶Raviart-Thomas混合有限元离散状态和对偶状态变量,利用分片线性函数逼近控制变量,获得了状态变量和控制变量的最大模误差估计,这里控制变量的收敛阶是h~2,状态变量的收敛阶是h3/2|lnh|1/2.最后利用数值算例验证了理论结果. 相似文献
14.
Eduardo M. Garau Pedro Morin 《Numerical Methods for Partial Differential Equations》2017,33(4):1266-1282
We develop an a posteriori error estimator which focuses on the local H1 error on a region of interest. The estimator bounds a weighted Sobolev norm of the error and is efficient up to oscillation terms. The new idea is very simple and applies to a large class of problems. An adaptive method guided by this estimator is implemented and compared to other local estimators, showing an excellent performance. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1266–1282, 2017 相似文献
15.
In this article, we construct an a posteriori error estimator for expanded mixed hybrid finite‐element methods for second‐order elliptic problems. An a posteriori error analysis yields reliable and efficient estimate based on residuals. Several numerical examples are presented to show the effectivity of our error indicators. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 330–349, 2007 相似文献
16.
In this article, we construct and analyze a residual-based a posteriori error estimator for a quadratic finite volume method (FVM) for solving nonlinear elliptic partial differential equations with homogeneous Dirichlet boundary conditions. We shall prove that the a posteriori error estimator yields the global upper and local lower bounds for the norm error of the FVM. So that the a posteriori error estimator is equivalent to the true error in a certain sense. Numerical experiments are performed to illustrate the theoretical results. 相似文献
17.
In this article, we propose a new discontinuous finite volume element (DFVE) method for the second‐order elliptic problems. We treat the DFVE method as a perturbation of the interior penalty method and get a superapproximation estimate in a mesh dependent norm between the solution of the DFVE method and that of the interior penalty method. This reveals that the DFVE method is much closer to the interior penalty method than we have known. By using this superapproximation estimate, we can easily get the optimal order error estimates in the L2 ‐norm and in the maximum norms of the DFVE method.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 425–440, 2012 相似文献
18.
We use Galerkin least-squares terms and biorthogonal wavelet bases to develop a new stabilized dual-mixed finite element method for second-order elliptic equations in divergence form with Neumann boundary conditions. The approach introduces the trace of the solution on the boundary as a new unknown that acts also as a Lagrange multiplier. We show that the resulting stabilized dual-mixed variational formulation and the associated discrete scheme defined with Raviart–Thomas spaces are well-posed and derive the usual a priori error estimates and the corresponding rate of convergence. Furthermore, a reliable and efficient residual-based a posteriori error estimator and a reliable and quasi-efficient one are provided. 相似文献
19.
20.
Georg Dolzmann. 《Mathematics of Computation》1999,68(228):1397-1427
We prove a priori estimates and optimal error estimates for linear finite element approximations of elliptic systems in divergence form with continuous coefficients in Campanato spaces. The proofs rely on discrete analogues of the Campanato inequalities for the solution of the system, which locally measure the decay of the energy. As an application of our results we derive -estimates and give a new proof of the well-known -results of Rannacher and Scott.