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1.
Error Estimates of Mixed Methods for Optimal Control Problems Governed by General Elliptic Equations
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Tianliang Hou & Li Li 《advances in applied mathematics and mechanics.》2016,8(6):1050-1071
In this paper, we investigate the error estimates of mixed finite element
methods for optimal control problems governed by general elliptic equations. The
state and co-state are approximated by the lowest order Raviart-Thomas mixed finite
element spaces and the control variable is approximated by piecewise constant functions.
We derive $L^2$ and $H^{-1}$-error estimates both for the control variable and the state
variables. Finally, a numerical example is given to demonstrate the theoretical results. 相似文献
2.
Error Estimates and Superconvergence of Mixed Finite Element Methods for Optimal Control Problems with Low Regularity
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Yanping Chen Tianliang Hou & Weishan Zheng 《advances in applied mathematics and mechanics.》2012,4(6):751-768
In this paper, we investigate the error estimates and
superconvergence property of mixed finite element methods for
elliptic optimal control problems. The state and co-state are
approximated by the lowest order Raviart-Thomas mixed finite element
spaces and the control variable is approximated by piecewise
constant functions. We derive $L^2$ and $L^\infty$-error
estimates for the control variable. Moreover, using a recovery
operator, we also derive some superconvergence results for the
control variable. Finally, a numerical example is given to
demonstrate the theoretical results. 相似文献
3.
A Posteriori Error Estimates of Triangular Mixed Finite Element Methods for Semilinear Optimal Control Problems
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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. 相似文献
4.
We consider the tensorial diffusion equation, and address the discrete maximum–minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum–minimum principle) of mixed finite element formulations. It is well-known that the classical finite element formulations (like the single-field Galerkin formulation, and Raviart–Thomas, variational multiscale, and Galerkin/least-squares mixed formulations) do not produce non-negative solutions (that is, they do not satisfy the discrete maximum–minimum principle) on arbitrary meshes and for strongly anisotropic diffusivity coefficients. 相似文献
5.
J.P. Pontaza J.N. Reddy 《Journal of Quantitative Spectroscopy & Radiative Transfer》2005,95(3):387-406
We present least-squares-based finite element formulations for the numerical solution of the radiative transfer equation in its first-order primitive variable form. The use of least-squares principles leads to a variational unconstrained minimization problem in a setting of residual minimization. In addition, the resulting linear algebraic problem will always have a symmetric positive definite coefficient matrix, allowing the use of robust and fast iterative methods for its solution. We consider space-angle coupled and decoupled formulations. In the coupled formulation, the space-angle dependency is represented by two-dimensional finite element expansions and the least-squares functional minimized in the continuous space-angle domain. In the decoupled formulation the angular domain is represented by discrete ordinates, the spatial dependence represented by one-dimensional finite element expansions, and the least-squares functional minimized continuously in space domain and at discrete locations in the angle domain. Numerical examples are presented to demonstrate the merits of the formulations in slab geometry, for absorbing, emitting, anisotropically scattering mediums, allowing for spatially varying absorption and scattering coefficients. For smooth solutions in space-angle domain, exponentially fast decay of error measures is demonstrated as the p-level of the finite element expansions is increased. The formulations represent attractive alternatives to weak form Galerkin finite element formulations, typically applied to the more complicated second-order even- and odd-parity forms of the radiative transfer equation. 相似文献
6.
7.
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. 相似文献
8.
A Mixed Finite Element Method for Stationary Magneto-Heat Coupling System with Variable Coefficients
In this article, a mixed finite element method for thermally coupled, stationary incompressible MHD problems with physical parameters dependent on temperature in the Lipschitz domain is considered. Due to the variable coefficients of the MHD model, the nonlinearity of the system is increased. A stationary discrete scheme based on the coefficients dependent temperature is proposed, in which the magnetic equation is approximated by Nédélec edge elements, and the thermal and Navier–Stokes equations are approximated by the mixed finite elements. We rigorously establish the optimal error estimates for velocity, pressure, temperature, magnetic induction and Lagrange multiplier with the hypothesis of a low regularity for the exact solution. Finally, a numerical experiment is provided to illustrate the performance and convergence rates of our numerical scheme. 相似文献
9.
基于对偶混合变分原理的Signorini问题的数值模拟 总被引:1,自引:0,他引:1
基于Signorini问题的对偶混合变分形式,提出了一种非协调有限元逼近格式,证明了离散的B-B条件,获得了Raviart-Thomas(k=0)有限元逼近的误差界O(h3/4),并且Uzawa型算法对协调与非协调有限元逼近格式进行了数值求解.根据数值结果的分析和比较,表明应用非协调有限元逼近格式求解更有效. 相似文献
10.
An approximation for the boundary optimal control problem of a heat equation defined in a variable domain
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In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of theboundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming (SQP) algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method. 相似文献
11.
Asymptotic Expansions and Extrapolations of $H^1$-Galerkin Mixed Finite Element Method for Strongly Damped Wave Equation
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In this paper, a high-accuracy $H^1$-Galerkin mixed finite element method (MFEM)
for strongly damped wave equation is studied by linear triangular finite element.
By constructing a suitable extrapolation scheme, the convergence rates can be improved
from $\mathcal{O}(h)$ to $\mathcal{O}(h^3)$ both for the original variable $u$ in $H^1(Ω)$ norm and for
the actual stress variable $\boldsymbol{P}=∇u_t$ in $H$(div;$Ω$) norm, respectively. Finally, numerical
results are presented to confirm the validity of the theoretical analysis and excellent
performance of the proposed method. 相似文献
12.
Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation
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We present the finite difference/element method for a
two-dimensional modified fractional diffusion equation. The analysis
is carried out first for the time semi-discrete scheme, and then for
the full discrete scheme. The time discretization is based on the
$L1$-approximation for the fractional derivative terms and the
second-order backward differentiation formula for the classical
first order derivative term. We use finite element method for the
spatial approximation in full discrete scheme. We show that both the
semi-discrete and full discrete schemes are unconditionally stable
and convergent. Moreover, the optimal convergence rate is obtained.
Finally, some numerical examples are tested in the case of one and
two space dimensions and the numerical results confirm our
theoretical analysis. 相似文献
13.
L. Beiro da Veiga V. Gyrya K. Lipnikov G. Manzini 《Journal of computational physics》2009,228(19):7215-7232
Various approaches to extend finite element methods to non-traditional elements (general polygons, pyramids, polyhedra, etc.) have been developed over the last decade. The construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis. The mimetic finite difference (MFD) method works on general polygonal meshes and has many similarities with low-order finite element methods. Both schemes try to preserve the fundamental properties of the underlying physical and mathematical models. The essential difference between the two schemes is that the MFD method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices. Since no extension of basis functions inside the mesh elements is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we present a new MFD method for the Stokes problem on arbitrary polygonal meshes and analyze its stability. The method is developed for the general case of tensor coefficients, which allows us to apply it to a linear elasticity problem, as well. Numerical experiments show, for the velocity variable, second-order convergence in a discrete L2 norm and first-order convergence in a discrete H1 norm. For the pressure variable, first-order convergence is shown in the L2 norm. 相似文献
14.
推导耦合过渡区内参变量信息交换的元/网格动量传递多尺度算法,建立离散元与有限元耦合时空多尺度计算模型,并应用于激光辐照下受拉铝板破坏行为的数值模拟中.通过对比有限元计算模型、空间多尺度计算模型与时空多尺度计算模型在激光辐照下受拉铝板破坏算例的模拟结果,验证离散元与有限元耦合时空多尺度计算模型的准确性和数值计算高效率优势.使用该多尺度计算模型从宏观和细观尺度对铝板破坏行为进行数值模拟,模拟结果与实验结果基本一致. 相似文献
15.
离散元与有限元结合的多尺度方法及其应用 总被引:11,自引:0,他引:11
在深入研究复杂结构和非均质材料冲击响应和破坏机理的过程中,往往遇到多尺度计算问题.提出并建立起离散元与有限元结合的多尺度方法,该方法采用离散元对感兴趣的局部进行细观尺度的模拟,利用有限元进行宏观的模拟,从而节约了计算时间.采用一种特殊的过渡层衔接离散元区和有限元区.将这一方法应用于激光辐照下预应力铝板的破坏响应,并将得到的模拟结果与实验进行了比较. 相似文献
16.
S. Hysing 《Journal of computational physics》2012,231(6):2449-2465
A new realization of a finite element level set method for simulation of immiscible fluid flows is introduced and validated on numerical benchmarks. The new method involves a mixed discretization of the dependent variables, discretizing the flow variables with non-conforming Rannacher–Turek finite elements while using a simple first order conforming discretization of the level set field. A three step segregated solution approach is employed, first a discrete projection method is used to decouple and compute the velocity and pressure separately, after which the level set field can be computed independently.The developed method is tested and validated on a static bubble test case and on a numerical rising bubble test case for which a very accurate benchmark solution has been established. The new approach is also compared against two commercial simulation codes, Ansys Fluent and Comsol Multiphysics, which shows that the developed method is a magnitude or more accurate and at the same time significantly faster than state of the art commercial codes. 相似文献
17.
针对薄板弯曲大变形问题, 运用变分原理, 建立了薄板弯曲大变形问题的高阶非线性偏微分方程. 运用有限差分法和动态设计变量优化算法原理, 以离散坐标点的上未知挠度为设计变量, 以离散坐标点的差分方程组构建目标函数, 提出了薄板弯曲大变形挠度求解的动态设计变量优化算法, 编制了相应的优化求解程序. 分析了具有固定边界、均布载荷下的矩形薄板挠度的典型算例. 通过与有限元的结果对比, 表明了本文求解算法的有效性和精确性, 提供了直接求解实际工程问题的基础. 相似文献
18.
In this paper, the streamline upwind/Petrov Galerkin (SUPG) stabilized virtual element method (VEM) for optimal control problem governed by a convection dominated diffusion equation is investigated. The virtual element discrete scheme is constructed based on the first-optimize-then-discretize strategy and SUPG stabilized virtual element approximation of the state equation and adjoint state equation. An a priori error estimate is derived for both the state, adjoint state, and the control. Numerical experiments are carried out to illustrate the theoretical findings. 相似文献
19.
求解辐射传递的非结构混合有限体积/有限元法 总被引:1,自引:0,他引:1
本文给了一种适用于任意非结构网格的有限体积/有限元法的混合算法用于求解多维半透明吸收、发射、散射性灰矩形介质内的辐射传递.该方法使用有限元法进行角度离散,有限体积法进行空间离散.与基于辐射传递离散坐标方程的方法不同的是,该方法在迭代求解的过程中,针对每一个空间体元,所有角度方向的辐射强度同时耦合求出.通过两个算例验证了该解法的正确性. 相似文献