共查询到20条相似文献，搜索用时 15 毫秒
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《Numerical Methods for Partial Differential Equations》2018,34(1):336-356
In this article, we study the superconvergence analysis of conforming bilinear finite element method (FEM) for nonlinear Joule heating equations. Based on the rigorous estimates together with high accuracy analysis of this element, mean value technique and interpolation postprocessing approach, the superclose and superconvergent estimates about the related variables in H1‐norm are derived for semidiscrete and a linearized backward Euler fully discrete schemes, which extends the results of optimal estimates obtained for conforming FEMs in the previous literature. At last, a numerical experiment is performed to verify the theoretical analysis. 相似文献
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Bhupen Deka Tazuddin Ahmed 《Numerical Methods for Partial Differential Equations》2013,29(5):1522-1542
A finite element method is proposed and analyzed for hyperbolic problems with discontinuous coefficients. The main emphasize is given on the convergence of such method. Due to low global regularity of the solutions, the error analysis of the standard finite element method is difficult to adopt for such problems. For a practical finite element discretization, optimal error estimates in L∞(L2) and L∞(H1) norms are established for continuous time discretization. Further, a fully discrete scheme based on a symmetric difference approximation is considered, and optimal order convergence in L∞(H1) norm is established. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
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In this article, we consider the semidiscrete and the backward Euler fully discrete discontinuous finite volume element methods for the second‐order parabolic problems and obtain the optimal order error estimates in a mesh dependent norm and in the L2‐norm. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 相似文献
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Dongyang Shi Huaijun Yang 《Numerical Methods for Partial Differential Equations》2019,35(3):1206-1223
This article concerns with the superconvergence analysis of bilinear finite element method (FEM) for nonlinear Poisson–Nernst–Planck (PNP) equations. By employing high accuracy integral identities together with mean value technique, the superclose estimates in H1‐norm are derived for the semi‐discrete and the backward Euler fully‐discrete schemes, which improve the suboptimal error estimate in L2‐norm in the previous literature. Furthermore, the global superconvergence results in H1‐norm are obtained through interpolation postprocessing approach. Finally, a numerical example is provided to confirm the theoretical analysis. 相似文献
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The conforming bilinear mixed finite element method is established for a fourth‐order wave equation. By applying the interpolation and projection simultaneously, the superclose and superconvergence results of order O (h 2) for the original variable u and intermediate variable p =? a 1(X )Δu in H1‐norm are achieved for the semi‐discrete and fully discrete schemes, respectively (where a 1(X ) is the coefficient appeared in the equation). The distinct advantage of this method is that it can reduce the smoothness of solutions u ,u t , and p compared with the existing literature for getting optimal estimates alone. At last, some numerical results are carried out to validate the theoretical analysis. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
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《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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Tingchun Wang 《Numerical Methods for Partial Differential Equations》2017,33(3):840-867
In this article, a decoupled and linearized compact finite difference scheme is proposed for solving the coupled nonlinear Schrödinger equations. The new scheme is proved to preserve the total mass and energy which are defined by using a recursion relationship. Besides the standard energy method, an induction argument together with an H1 technique are introduced to establish the optimal point‐wise error estimate of the proposed scheme. Without imposing any constraints on the grid ratios, the convergence order of the numerical solution is proved to be of with mesh size h and time step τ. Numerical results are reported to verify the theoretical analysis, and collision of two solitary waves are also simulated. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 840–867, 2017 相似文献
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在各向异性网格下,针对具有Caputo导数的二维多项时间分数阶扩散方程,给出了线性三角形元的高精度分析.首先,基于线性三角形元和改进的L1格式,建立了一个全离散逼近格式,并证明了其无条件稳定性;其次,利用有限元插值算子与Riesz投影算子之间的关系及相关的高精度结果,导出了超逼近性质.进而,借助于插值后处理技术得到了超收敛估计.值得指出的是,单独利用插值算子或Riesz投影都无法得到上述超逼近和超收敛结果.最后,利用数值算例验证了理论分析的正确性.此外,对一些常见的有限单元在该方程的数值逼近方面,作了进一步探讨. 相似文献
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《Numerical Methods for Partial Differential Equations》2018,34(2):705-730
In this article, an efficient algorithm for the evaluation of the Caputo fractional derivative and the superconvergence property of fully discrete finite element approximation for the time fractional subdiffusion equation are considered. First, the space semidiscrete finite element approximation scheme for the constant coefficient problem is derived and supercloseness result is proved. The time discretization is based on the L1‐type formula, whereas the space discretization is done using, the fully discrete scheme is developed. Under some regularity assumptions, the superconvergence estimate is proposed and analyzed. Then, extension to the case of variable coefficients is also discussed. To reduce the computational cost, the fast evaluation scheme of the Caputo fractional derivative to solve the fractional diffusion equations is designed. Finally, numerical experiments are presented to support the theoretical results. 相似文献
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** Email: silvia{at}mat.uc.pt*** Email: ferreira{at}mat.uc.pt**** Email: grigo{at}math.tu-berlin.de In this paper we study the convergence of a centred finite differencescheme on a non-uniform mesh for a 1D elliptic problem subjectto general boundary conditions. On a non-uniform mesh, the schemeis, in general, only first-order consistent. Nevertheless, weprove for s (1/2, 2] order O(hs)-convergence of solution andgradient if the exact solution is in the Sobolev space H1+s(0,L), i.e. the so-called supraconvergence of the method. It isshown that the scheme is equivalent to a fully discrete linearfinite-element method and the obtained convergence order isthen a superconvergence result for the gradient. Numerical examplesillustrate the performance of the method and support the convergenceresult. 相似文献
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Guanying Sun Dong Liang Wenqia Wang 《Numerical Methods for Partial Differential Equations》2009,25(2):470-493
In this article we consider the age structured population growth model of marine invertebrates. The problem is a nonlinear coupled system of the age‐density distribution of sessile adults and the abundance of larvae. We propose the semidiscrete and fully‐discrete discontinuous Galerkin schemes to the nonlinear problem. The DG method is well suited to approximate the local behavior of the problem and to easily take the locally refined meshes with hanging nodes adaptively. The simple communication pattern between elements makes the DG method ideal for parallel computation. The global existence of the approximation solution is proved for the nonlinear approximation system by using the broken Sobolev spaces and the Schauder's fixed point theorem, and error estimates are obtained for both the semidiscrete scheme and the fully‐discrete scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 相似文献
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Gujji Murali Mohan Reddy Rajen Kumar Sinha 《Numerical Methods for Partial Differential Equations》2016,32(5):1309-1330
We derive two optimal a posteriori error estimators for an implicit fully discrete approximation to the solutions of linear integro‐differential equations of the parabolic type. A continuous, piecewise linear finite element space is used for the space discretization and the time discretization is based on an implicit backward Euler method. The a posteriori error indicator corresponding to space discretization is derived using the anisotropic interpolation estimates in conjunction with a Zienkiewicz‐Zhu error estimator to approach the error gradient. The error due to time discretization is derived using continuous, piecewise linear polynomial in time. We use the linear approximation of the Volterra integral term to estimate the quadrature error in the second estimator. Numerical experiments are performed on the isotropic mesh to validate the derived results.© 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1309–1330, 2016 相似文献
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采用双线性元及零阶Raviart-Thomas元（Q11+Q10×Q01）对非线性抛物方程讨论了一种H1-Galerkin混合有限元方法.提出一个线性化的二阶格式,利用数学归纳法有技巧的导出了原始变量u在H1（Ω）模意义下及流量p=▽u在L2（Ω）模意义下的O（h2+τ2）阶超逼近性质.引入一个有关初始点的时间离散方程,并利用其得到了▽ ·在L2（Ω）模意义下的O（h2+τ2）阶的超逼近结果.同时利用插值后处理技巧得到整体超收敛.最后,数值算例结果验证了理论分析（其中,h是剖分参数,τ是时间步长）. 相似文献
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Graeme Fairweather Heping Ma Weiwei Sun 《Numerical Methods for Partial Differential Equations》2008,24(2):449-464
An orthogonal spline collocation (OSC) spatial discretization is proposed for the solution of the fully coupled stream function‐vorticity formulation of the Navier–Stokes equations in two dimensions. For the time‐stepping, a three‐level leapfrog scheme is employed. This method is algebraically linear, and, at each time step, gives rise to a system of linear equations of the form arising in the OSC approximation of the biharmonic Dirichlet problem and can be solved by a fast direct method. Error estimates in the Hl–norm in space, l = 1,2, are derived for the semi‐discrete method and the fully‐discrete leapfrog scheme which is also shown to be second order accurate in time. Numerical results are presented which confirm the theoretical analysis and exhibit superconvergence phenomena, which provide superconvergent approximations to the components of the velocity. © John Wiley & Sons, Inc. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 相似文献
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Hang Chen;Yuyu He;Hongtao Chen; 《Numerical Methods for Partial Differential Equations》2024,40(1):e23067
In this article, we construct first- and second-order semidiscrete schemes for the magnetohydrodynamics (MHD) equations with variable density based on scalar auxiliary variable (SAV) approach. These schemes are decoupled, unconditionally energy stable and only solve a sequence of linear differential equations at each time step. We carry out a rigorous error analysis for the first-order SAV scheme in two-dimensional case. Some numerical experiments are presented to verify the accuracy and stability. 相似文献
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Gatica Gabriel N.; Heuer Norbert; Meddahi Salim 《IMA Journal of Numerical Analysis》2003,23(2):301-330
We provide a general abstract theory for the solvability andGalerkin approximation of nonlinear twofold saddle point problems.In particular, a Strang error estimate containing the consistencyterms arising from the approximation of the continuous operatorsinvolved is deduced. Then we apply these results to analysea fully discrete Galerkin scheme for a twofold saddle pointformulation of a nonlinear elliptic boundary value problem indivergence form. Some numerical results are also presented. 相似文献
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A linearized three‐step backward differential formula (BDF) Galerkin finite element method (FEM) is developed for nonlinear Sobolev equation with bilinear element. Temporal error and spatial error are discussed through introducing a time‐discrete system. Solutions of the time‐discrete system are bounded in H2‐norm by the temporal error. Superconvergence results of order O(h2 + τ3) in H1‐norm for the original variable are deduced based on the spatial error. Some new tricks are utilized to get higher order of the temporal error and the spatial error. At last, two numerical examples are provided to support the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step. 相似文献
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We consider the wave equation, on a multidimensional spatial domain. The discretization of the spatial domain is performed using a general class of nonconforming meshes which has been recently studied for stationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMAJ Numer Anal 30 (2010), 1009–1043). The discretization in time is performed using a uniform mesh. We derive a new implicit finite volume scheme approximating the wave equation and we prove error estimates of the finite volume approximate solution in several norms which allow us to derive error estimates for the approximations of the exact solution and its first derivatives. We prove in particular, when the discrete flux is calculated using a stabilized discrete gradient, the convergence order is begin{align*} h_mathcal{D}end{align*} (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumption begin{align*}uin C^3(lbrack 0,Trbrack;C^2(overline{Omega}))end{align*} for the exact solution u. The proof of these error estimates is based essentially on a comparison between the finite volume approximate solution and an auxiliary finite volume approximation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献