The a Posteriori Error Estimates for Chebyshev-Galerkin Spectral Methods in One Dimension

In this paper, the Chebyshev-Galerkin spectral approximations are employed to investigate Poisson equations and the fourth order equations in one dimension. Meanwhile, $p$-version finite element methods with Chebyshev polynomials are utilized to solve Poisson equations. The efficient and reliable a posteriori error estimators are given for different models. Furthermore, the a priori error estimators are derived independently. Some numerical experiments are performed to verify the theoretical analysis for the a posteriori error indicators and a priori error estimations.     

A Posteriori Error Estimates of Triangular Mixed Finite Element Methods for Semilinear Optimal Control Problems

In this paper, we present an a posteriori error estimates of semilinear quadratic constrained optimal control problems using triangular mixed finite element methods. The state and co-state are approximated by the order $k\leq 1$ Raviart- Thomas mixed finite element spaces and the control is approximated by piecewise constant element. We derive a posteriori error estimates for the coupled state and control approximations. A numerical example is presented in confirmation of the theory.     

A Priori and a Posteriori Error Analysis of the Discontinuous Galerkin Methods for Reissner-Mindlin Plates

In this paper, we apply an a posteriori error control theory that we develop in a very recent paper to three families of the discontinuous Galerkin methods for the Reissner-Mindlin plate problem. We derive robust a posteriori error estimators for them and prove their reliability and efficiency.     

非匹配网格上广义Stokes问题的区域分裂解法及事后误差估算

发展了一种广义Stokes问题的无覆盖区域分裂解法。子域交界面上的约束条件是通过引入一Lagrange乘子而得到弱满足的,在有限元离散子域的交界处网格可以是非匹配的。应用Petrov Galerkin方法解每个子域上的广义Stokes问题,而交界面上的Lagrange乘子则通过共轭梯度法迭代求解,各变量均由线性函数离散。对上述区域分裂解法,还构造了基于求解当地问题的误差事后估算方法。各变量的当地误差估算器定义在二阶非连续鼓包(bump)函数的空间中。最后给出了基于事后误差估算值的自适应网格上的数值结果。     

自适应有限元方法和后验误差估计

自适应有限元方法是科学研究和工程设计领域中非常有效的一种求解偏微分方程的数值计算方法。这种方法是为了以尽可能小的代价取得尽可能好的计算效果。后验误差估计是实现自适应有限元计算的关键性手段。文章综合介绍了自适应有限元方法和后验误差估计在求解椭圆型方程、抛物型方程和双曲型方程方面所取得的比较新的成就。     

Error Analysis and Adaptive Methods of Least Squares Nonconforming Finite Element for the Transport Equations

In this paper, we consider a least squares nonconforming finite element of low order for solving the transport equations. We give a detailed overview on the stability and the convergence properties of our considered methods in the stability norm. Moreover, we derive residual type a posteriori error estimates for the least squares nonconforming finite element methods under $H^{−1}$-norm, which can be used as the error indicators to guide the mesh refinement procedure in the adaptive finite element method. The theoretical results are supported by a series of numerical experiments.     

Non-uniform order mixed FEM approximation: Implementation,post-processing,computable error bound and adaptivity

The present work provides a straightforward and focused set of tools and corresponding theoretical support for the implementation of an adaptive high order finite element code with guaranteed error control for the approximation of elliptic problems in mixed form. The work contains: details of the discretisation using non-uniform order mixed finite elements of arbitrarily high order; a new local post-processing scheme for the primary variable; the use of the post-processing scheme in the derivation of new, fully computable bounds for the error in the flux variable; and, an hp-adaptive refinement strategy based on the a posteriori error estimator. Numerical examples are presented illustrating the results obtained when the procedure is applied to a challenging problem involving a ten-pole electric motor with singularities arising from both geometric features and discontinuities in material properties. The procedure is shown to be capable of producing high accuracy numerical approximations with relatively modest numbers of unknowns.     

计算圆柱绕流拉力系数的自适应迎风有限元方法

研究圆柱绕流中拉力系数的自适应计算问题.对二维定常粘性不可压Navier-Stokes方程使用迎风有限元格式,通过引入对偶问题,得到圆柱拉力系数的加权后验误差估计器,实现了自适应网格加密.数值实验表明格式的可行性以及估计器的稳健性.     

A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on hierarchical a posteriori error estimates. A global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution. Instead of solving the global error problem exactly, which is costly in general, we solve it iteratively using the symmetric Gauß–Seidel method. Numerical results show that a few GS iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation. The new method is compared with several strategies using local error estimators or recovered Hessians. Numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients.     

二维非线性抛物方程广义差分法/有限体积法的自适应计算

给出三角网上二维非线性抛物方程广义差分法(有限体积法)的一种基于残量估计的后验误差估计,并在此基础上设计了自适应计算方案,以适应物理解在时空的大梯度变化.提出了适合发展方程自适应计算的三角网数据结构(不是树状结构)和灵活的局部粗化算法.     

Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes

We developed a new monotone finite volume method for diffusion equations. The second-order linear methods, such as the multipoint flux approximation, mixed finite element and mimetic finite difference methods, are not monotone on strongly anisotropic meshes or for diffusion problems with strongly anisotropic coefficients. The finite volume (FV) method with linear two-point flux approximation is monotone but not even first-order accurate in these cases. The developed monotone method is based on a nonlinear two-point flux approximation. It does not require any interpolation scheme and thus differs from other nonlinear finite volume methods based on a two-point flux approximation. The second-order convergence rate is verified with numerical experiments.     

Unified a Priori Error Estimate and a Posteriori Error Estimate of CIP-FEM for Elliptic Equations

This paper is devoted to a unified a priori and a posteriori error analysis of CIP-FEM (continuous interior penalty finite element method) for second-order elliptic problems. Compared with the classic a priori error analysis in literature, our technique can easily apply for any type regularity assumption on the exact solution, especially for the case of lower $H^{1+s}$ weak regularity under consideration, where 0 ≤$s$≤ 1/2. Because of the penalty term used in the CIP-FEM, Galerkin orthogonality is lost and Céa Lemma for conforming finite element methods can not be applied immediately when 0≤$s$≤1/2. To overcome this difficulty, our main idea is introducing an auxiliary $C^1$ finite element space in the analysis of the penalty term. The same tool is also utilized in the explicit a posteriori error analysis of CIP-FEM.     

自适应间断有限元方法求解双曲守恒律方程

研究自适应Runge-Kutta间断Galerkin (RKDG)方法求解双曲守恒律方程组,并提出两种生成相容三角形网格的自适应算法.第一种算法适用于规则网格,实现简单、计算速度快.第二种算法基于非结构网格,设计一类基于间断界面的自适应网格加密策略,方法灵活高效.两种方法都具有令人满意的计算效果,而且降低了RKDG的计算量.     

Standard and goal-oriented adaptive mesh refinement applied to radiation transport on 2D unstructured triangular meshes

Standard and goal-oriented adaptive mesh refinement (AMR) techniques are presented for the linear Boltzmann transport equation. A posteriori error estimates are employed to drive the AMR process and are based on angular-moment information rather than on directional information, leading to direction-independent adapted meshes. An error estimate based on a two-mesh approach and a jump-based error indicator are compared for various test problems. In addition to the standard AMR approach, where the global error in the solution is diminished, a goal-oriented AMR procedure is devised and aims at reducing the error in user-specified quantities of interest. The quantities of interest are functionals of the solution and may include, for instance, point-wise flux values or average reaction rates in a subdomain. A high-order (up to order 4) Discontinuous Galerkin technique with standard upwinding is employed for the spatial discretization; the discrete ordinates method is used to treat the angular variable.     

Heuristic a posteriori estimation of error due to dissipation in finite volume schemes and application to mesh adaptation

A heuristic method is proposed to estimate a posteriori that part of the total discretization error which is attributable to the smoothing effect of added dissipation, for finite volume discretizations of the Euler equations. This is achieved by observing variation in a functional of the solution as the level of dissipation is varied, and it is deduced for certain test-cases that the dissipation alone accounts for the majority of the functional error. Based on this result an error estimator and mesh adaptation indicator is proposed for functionals, relying on the solution of an adjoint problem. The scheme is considerably implementationally simpler and computationally cheaper than other recently proposed a posteriori error estimators for finite volume schemes, but does not account for all sources of error. In mind of this, emphasis is placed on numerical evaluation of the performance of the indicator, and it is shown to be extremely effective in both estimating and reducing error for a range of 2d and 3d flows.     

流动数值模拟中一种并行自适应有限元算法

给出了一种流动数值模拟中的基于误差估算的并行网格自适应有限元算法.首先,以初网格上获得的当地事后误差估算值为权,应用递归谱对剖分方法划分初网格,使各子域上总体误差近似相等,以解决负载平衡问题.然后以误差值为判据对各子域内网格进行独立的自适应处理.最后应用基于粘接元的区域分裂法在非匹配的网格上求解N-S方程.区域分裂情形下N-S方程有限元解的误差估算则是广义Stokes问题误差估算方法的推广.为验证方法的可靠性,给出了不可压流经典算例的数值结果.     

A multi-moment finite volume formulation for shallow water equations on unstructured mesh

A novel and accurate finite volume method has been presented to solve the shallow water equations on unstructured grid in plane geometry. In addition to the volume integrated average (VIA moment) for each mesh cell, the point values (PV moment) defined on cell boundary are also treated as the model variables. The volume integrated average is updated via a finite volume formulation, and thus is numerically conserved, while the point value is computed by a point-wise Riemann solver. The cell-wise local interpolation reconstruction is built based on both the VIA and the PV moments, which results in a scheme of almost third order accuracy. Efforts have also been made to formulate the source term of the bottom topography in a way to balance the numerical flux function to satisfy the so-called C-property. The proposed numerical model is validated by numerical tests in comparison with other methods reported in the literature.     

Mesh quality effects on the accuracy of CFD solutions on unstructured meshes

The order of accuracy and error magnitude of node- and cell-centered schemes are examined on representative unstructured meshes and flowfield solutions for computational fluid dynamics. Specifically, we investigate the properties of inviscid and viscous flux discretizations for isotropic and highly stretched meshes using the Method of Manufactured Solutions. Grid quality effects are studied by randomly perturbing the base meshes and cataloguing the error convergence as a function of grid size. For isotropic grids, node-centered approaches produce less error than cell-centered approaches. Moreover, a corrected node-centered scheme is shown to maintain third order accuracy for the inviscid terms on arbitrary triangular meshes. In contrast, for stretched meshes, cell-centered schemes are favored, with cell-centered prismatic approaches in particular showing the lowest levels of error. In three dimensions, simple flux integrations on non-planar control volume faces lead to first-order solution errors, while second-order accuracy is recovered by triangulation of the non-planar faces.     

FORCE schemes on unstructured meshes I: Conservative hyperbolic systems

This paper is about the construction of numerical fluxes of the centred type for one-step schemes in conservative form for solving general systems of conservation laws in multiple space dimensions on structured and unstructured meshes. The work is a multi-dimensional extension of the one-dimensional FORCE flux and is closely related to the work of Nessyahu–Tadmor and Arminjon. The resulting basic flux is first-order accurate and monotone; it is then extended to arbitrary order of accuracy in space and time on unstructured meshes in the framework of finite volume and discontinuous Galerkin methods. The performance of the schemes is assessed on a suite of test problems for the multi-dimensional Euler and Magnetohydrodynamics equations on unstructured meshes.     

高分五号卫星多角度偏振相机最优化估计反演:角度依赖与后验误差分析

多角度偏振相机(directional polarimetric camera, DPC)随高分五号卫星已经成功发射并持续获取全球观测数据.针对DPC在陆地气溶胶反演领域的应用需求,本研究基于多参数最优化估计反演框架,引入信息量和后验误差分析工具,讨论了DPC观测信息量对角度的依赖,给出了地表和气溶胶参数的后验误差,并分析了后验误差的影响因素.研究表明:1)卫星观测信息量随观测角度个数的增加显著提升, DPC多角度观测比单角度观测的总DFS(degree of freedom for signal)平均提高了5.45; 2)气溶胶反演比地表更依赖于卫星观测几何,散射角覆盖范围越大,观测包含的气溶胶信息量越多; 3)反演参数的后验误差随观测角度个数的增加显著降低,而气溶胶模型误差对后验误差的影响并不显著.总体来说,观测误差是影响反演结果不确定性的主要因素.本研究对DPC多角度偏振观测的反演能力以及反演不确定性进行了系统的定量评估,为DPC在轨测试及反演算法开发提供参考.