Gradient Recovery Based Two-Grid Finite Element Method for Parabolic Integro-Differential Optimal Control Problems

In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and co-state variables, and piecewise constant function is used to approximate control variables. Generally, the optimal conditions for the problem are solved iteratively until the control variable reaches error tolerance. In order to calculate all the variables individually and parallelly, we introduce a gradient recovery based two-grid method. First, we solve the small scaled optimal control problem on coarse grids. Next, we use the gradient recovery technique to recover the gradients of state and co-state variables. Finally, using the recovered variables, we solve the large scaled optimal control problem for all variables independently. Moreover, we estimate priori error for the proposed scheme, and use an example to validate the theoretical results.

SUPERCONVERGENCE OF GRADIENT RECOVERY SCHEMES ON GRADED MESHES FOR CORNER SINGULARITIES

For the linear finite element solution to the Poisson equation, we show that supercon- vergence exists for a type of graded meshes for corner singularities in polygonal domains. In particular, we prove that the L^2-projection from the piecewise constant field △↓UN to the continuous and piecewise linear finite element space gives a better approximation of △↓U in the Hi-norm. In contrast to the existing superconvergence results, we do not assume high regularity of the exact solution.

SomeWeighted Averaging Methods for Gradient Recovery

We propose some new weighted averaging methods for gradient recovery,and present analytical and numerical investigation on the performance of these weighted averaging methods.It is shown analytically that the harmonic averaging yields a superconvergent gradient for any mesh in one-dimension and the rectangular mesh in two-dimension.Numerical results indicate that these new weighted averaging methods are better recovered gradient approaches than the simple averaging and geometry averaging methods under triangular mesh.

Gradient Recovery-Type a Posteriori Error Estimates for Steady-State Poisson-Nernst-Planck Equations

In this article,we derive the a posteriori error estimators for a class of steadystate Poisson-Nernst-Planck equations.Using the gradient recovery operator,the upper and lower bounds of the a posteriori error estimators are established both for the electrostatic potential and concentrations.It is shown by theory and numerical experiments that the error estimators are reliable and the associated adaptive computation is efficient for the steady-state PNP systems.

A Modified Polynomial Preserving Recovery and Its Applications to A Posteriori Error Estimates

A modified polynomial preserving gradient recovery technique is proposed. Unlike the polynomial preserving gradient recovery technique,the gradient recovered with the modified polynomial preserving recovery(MPPR) is constructed element-wise, and it is discontinuous across the interior edges.One advantage of the MPPR technique is that the implementation is easier when adaptive meshes are involved.Superconvergence results of the gradient recovered with MPPR are proved for finite element methods for elliptic boundary problems and eigenvalue problems under adaptive meshes. The MPPR is applied to adaptive finite element methods to construct asymptotic exact a posteriori error estimates.Numerical tests are provided to examine the theoretical results and the effectiveness of the adaptive finite element algorithms.

Convergent Adaptive Finite Element Method Based on Centroidal Voronoi Tessellations and Superconvergence

We present a novel adaptive finite element method(AFEM)for elliptic equations which is based upon the Centroidal Voronoi Tessellation(CVT)and superconvergent gradient recovery.The constructions of CVT and its dual Centroidal Voronoi Delaunay Triangulation(CVDT)are facilitated by a localized Lloyd iteration to produce almost equilateral two dimensional meshes.Working with finite element solutions on such high quality triangulations,superconvergent recovery methods become particularly effective so that asymptotically exact a posteriori error estimations can be obtained.Through a seamless integration of these techniques,a convergent adaptive procedure is developed.As demonstrated by the numerical examples,the new AFEM is capable of solving a variety of model problems and has great potential in practical applications.

Retrieving Topological Information of Implicitly Represented Diffuse Interfaces with Adaptive Finite Element Discretization

We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework.Utilizing an adaptive finite element implementation with effective gradient recovery techniques,we discuss how the Euler number can be accurately computed directly from the numerically solved phase field functions or order parameters.Numerical examples and applications to the topological analysis of point clouds are also presented.

Study on Recovering the Earth＇s Potential Field Based on GOCE Gradiometry

Given the second radial derivative Vrr（P） ｜δs of the Earth＇s gravitational potential V（P） on the surface δS corresponding to the satellite altitude, by using the fictitious compress recovery method, a fictitious regular harmonic field rrVrr（P）^＊ and a fictitious second radial gradient field V：（P） in the domain outside an inner sphere Ki can be determined, which coincides with the real field V（P） in the domain outside the Earth. Vrr^＊（P）could be further expressed as a uniformly convergent expansion series in the domain outside the inner sphere, because rrV（P）^＊ could be expressed as a uniformly convergent spherical harmonic expansion series due to its regularity and harmony in that domain. In another aspect, the fictitious field V^＊（P） defined in the domain outside the inner sphere, which coincides with the real field V（P） in the domain outside the Earth, could be also expressed as a spherical harmonic expansion series. Then, the harmonic coefficients contained in the series expressing V^＊（P） can be determined, and consequently the real field V（P） is recovered. Preliminary simulation calculations show that the second radial gradient field Vrr（P） could be recovered based only on the second radial derivative V（P）｜δs given on the satellite boundary. Concerning the final recovery of the potential field V（P） based only on the boundary value Vrr （P）｜δs, the simulation tests are still in process.