A Priori Error Analysis for NCVEM Discretization of Elliptic Optimal Control Problem

In this paper, we propose the nonconforming virtual element method (NCVEM) discretization for the pointwise control constraint optimal control problem governed by elliptic equations. Based on the NCVEM approximation of state equation and the variational discretization of control variables, we construct a virtual element discrete scheme. For the state, adjoint state and control variable, we obtain the corresponding prior estimate in H1 and L2 norms. Finally, some numerical experiments are carried out to support the theoretical results.

A PRIORI ERROR ESTIMATES FOR OBSTACLE OPTIMAL CONTROL PROBLEM,WHERE THE OBSTACLE IS THE CONTROLITSELF

In this paper we deal with the convergence analysis of the finite element method for an elliptic penalized unilateral obstacle optimal control problem where the control and the obstacle coincide.Error estimates are established for both state and control variables.We apply a fixed point type iteration method to solve the discretized problem.To corroborate our error estimations and the eficiency of our algorithms,the convergence results and numerical experiments are illustrated by concrete examples.

A Priori Error Estimates for Spectral Galerkin Approximations of Integral State-Constrained Fractional Optimal Control Problems

The fractional optimal control problem leads to significantly increased computational complexity compared to the corresponding classical integer-order optimal control problem,due to the global properties of fractional differential operators.In this paper,we focus on an optimal control problem governed by fractional differential equations with an integral constraint on the state variable.By the proposed first-order optimality condition consisting of a Lagrange multiplier,we design a spectral Galerkin discrete scheme with weighted orthogonal Jacobi polynomials to approximate the resulting state and adjoint state equations.Furthermore,a priori error estimates for state,adjoint state and control variables are discussed in details.Illustrative numerical tests are given to demonstrate the validity and applicability of our proposed approximations and theoretical results.

A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.

A PRIORI ERROR ESTIMATES FOR LEAST-SQUARES MIXED FINITE ELEMENT APPROXIMATION OF ELLIPTIC OPTIMAL CONTROL PROBLEMS

In this paper, a constrained distributed optimal control problem governed by a first- order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-Brezzi consistency condition, are used for solving the elliptic system with two unknown state variables. By adopting the Lagrange multiplier approach, continuous and discrete optimality systems including a primal state equation, an adjoint state equation, and a variational inequality for the optimal control are derived, respectively. Both the discrete state equation and discrete adjoint state equation yield a symmetric and positive definite linear algebraic system. Thus, the popular solvers such as preconditioned conjugate gradient （PCG） and algebraic multi-grid （AMG） can be used for rapid solution. Optimal a priori error estimates are obtained, respectively, for the control function in L2 （Ω）-norm, for the original state and adjoint state in H1 （Ω）-norm, and for the flux state and adjoint flux state in H（div; Ω）-norm. Finally, we use one numerical example to validate the theoretical findings.

A Priori Error Estimates of Finite Element Methods for Linear Parabolic Integro-Differential Optimal Control Problems

In this paper,we study the mathematical formulation for an optimal control problem governed by a linear parabolic integro-differential equation and present the optimality conditions.We then set up its weak formulation and the finite element approximation scheme.Based on these we derive the a priori error estimates for its finite element approximation both in H1 and L^(2)norms.Furthermore some numerical tests are presented to verify the theoretical results.

Adaptive Finite Element Method Based on Optimal Error Estimates for Linear Elliptic Problems on Nonconvex Polygonal Domains

The subject of this work is to propose adaptive finite element methods based on an optimal maximum norm error control estimate.Using estimators of the local regularity of the unknown exact solution derived from computed approximate solutions,the proposed procedures are analyzed in detail for a non-trivial class of corner problems and shown to be efficient in the sense that they generate the correct type of refinement and lead to the desired control under consideration.

The A Priori and A Posteriori Error Estimates for Modified Interior Transmission Eigenvalue Problem in Inverse Scattering

In this paper,we discuss the conforming finite element method for a modified interior transmission eigenvalues problem.We present a complete theoretical analysis for the method,including the a priori and a posteriori error estimates.The theoretical analysis is conducted under the assumption of low regularity on the solution.We prove the reliability and efficiency of the a posteriori error estimators for eigenfunctions up to higher order terms,and we also analyze the reliability of estimators for eigenvalues.Finally,we report numerical experiments to show that our posteriori error estimator is effective and the approximations can reach the optimal convergence order.The numerical results also indicate that the conforming finite element eigenvalues approximate the exact ones from below,and there exists a monotonic relationship between the conforming finite element eigenvalues and the refractive index through numerical experiments.

Virtual Element Discretization of Optimal Control Problem Governed by Brinkman Equations

In this paper, we discuss virtual element method (VEM) approximation of optimal control problem governed by Brinkman equations with control constraints. Based on the polynomial projections and variational discretization of the control variable, we build up the virtual element discrete scheme of the optimal control problem and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L2 and H1 norm are derived. The theoretical findings are illustrated by the numerical experiments.

A PRIORI AND A POSTERIORI ERROR ESTIMATES OF A WEAKLY OVER-PENALIZED INTERIOR PENALTY METHOD FOR NON-SELF-ADJOINT AND INDEFINITE PROBLEMS

In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a residual-based a posteriori error estimator, which is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to validate our theoretical analysis.

ERROR ANALYSIS FOR PARABOLIC OPTIMAL CONTROL PROBLEMS WITH MEASURE DATA IN A NONCONVEX POLYGONAL DOMAIN

This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain.Such problems usually possess low regularity in the state variable due to the presence of measure data and the nonconvex nature of the domain.The low regularity of the solution allows the finite element approximations to converge at lower orders.We prove the existence,uniqueness and regularity results for the solution to the control problem satisfying the first order optimality condition.For our error analysis we have used piecewise linear elements for the approximation of the state and co-state variables,whereas piecewise constant functions are employed to approximate the control variable.The temporal discretization is based on the implicit Euler scheme.We derive both a priori and a posteriori error bounds for the state,control and co-state variables.Numerical experiments are performed to validate the theoretical rates of convergence.

A Priori and a Posteriori Error Estimates for H(div)-Elliptic Problemwith Interior Penalty Method

In this paper,we propose and analyze the interior penalty discontinuous Galerkin method for H(div)-elliptic problem.An optimal a priori error estimate in the energy norm is proved.In addition,a residual-based a posteriori error estimator is obtained.The estimator is proved to be both reliable and efficient in the energy norm.Some numerical testes are presented to demonstrate the effectiveness of our method.

Convergence and Superconvergence of the Local Discontinuous Galerkin Method for Semilinear Second‑Order Elliptic Problems on Cartesian Grids

This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin(LDG)method for two-dimensional semilinear second-order elliptic problems of the form−Δu=f(x,y,u)on Cartesian grids.By introducing special GaussRadau projections and using duality arguments,we obtain,under some suitable choice of numerical fuxes,the optimal convergence order in L2-norm of O(h^(p+1))for the LDG solution and its gradient,when tensor product polynomials of degree at most p and grid size h are used.Moreover,we prove that the LDG solutions are superconvergent with an order p+2 toward particular Gauss-Radau projections of the exact solutions.Finally,we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves(p+1)-th order superconvergence.Some numerical experiments are performed to illustrate the theoretical results.

Error estimates of triangular mixed finite element methods for quasilinear optimal control problems

The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results.

A COUPLED METHOD COMBINING CROUZEIX-RAVIART NONCONFORMING AND NODE CONFORMING FINITE ELEMENT SPACES FOR BOIT CONSOLIDATION MODEL

A mixed finite element method is presented for the Biot consolidation problem in poroe-lasticity.More precisely,the displacement is approximated by using the Crouzeix-Raviart nonconforming finite elements,while the fuid pressure is approximated by using the node conforming finite elements.The well-posedness of the fully discrete scheme is established,and a corresponding priori error estimate with optimal order in the energy norm is also derived.Numerical experiments are provided to validate the theoretical results.

OPTIMAL AND PRESSURE-INDEPENDENT L2 VELOCITY ERROR ESTIMATES FOR A MODIFIED CROUZEIX-RAVIART STOKES ELEMENT WITH BDM RECONSTRUCTIONS

Nearly all inf-sup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a priori estimates for the ve- locity error become pressure-dependent, while divergence-free mixed finite elements de- liver pressure-independent estimates. A recently introduced new variational crime using lowest-order Raviart-Thomas velocity reconstructions delivers a much more robust modi- fied Crouzeix-Raviart element, obeying an optimal pressure-independent discrete H1 ve- locity estimate. Refining this approach, a more sophisticated variational crime employing the lowest-order BDM element is proposed, which also allows proving an optimal pressure- independent L2 velocity error. Numerical examples confirm the analysis and demonstrate the improved robustness in the Navier-Stokes case.

Error Estimates of Some Numerical Atomic Orbitals in Molecular Simulations

Numerical atomic orbitals have been successfully used in molecular simulations as a basis set,which provides a nature,physical description of the electronic states and is suitable for ■(N)calculations based on the strictly localized property.This paper presents a numerical analysis for some simplified atomic orbitals,with polynomial-type and confined Hydrogen-like radial basis functions respectively.We give some a priori error estimates to understand why numerical atomic orbitals are computationally efficient in electronic structure calculations.

AN IMPROVED ERROR ANALYSIS FOR FINITE ELEMENT APPROXIMATION OF BIOLUMINESCENCE TOMOGRAPHY

This paper is concerned with an ill-posed problem which results from the area of molecular imaging and is known as BLT problem. Using Tikhonov regularization technique, a quadratic optimization problem can be formulated. We provide an improved error estimate for the finite element approximation of the regularized optimization problem. Some numerical examples are presented to demonstrate our theoretical results.

Higher Order Triangular Mixed Finite Element Methods for Semilinear Quadratic Optimal Control Problems

In this paper,we investigate a priori error estimates for the quadratic optimal control problems governed by semilinear elliptic partial differential equations using higher order triangular mixed finite element methods.The state and the co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k≥0).A priori error estimates for the mixed finite element approximation of semilinear control problems are obtained.Finally,we present some numerical examples which confirm our theoretical results.

HIGH ORDER LOCAL DISCONTINUOUS GALERKIN METHODS FOR THE ALLEN-CAHN EQUATION： ANALYSIS AND SIMULATION

In this paper, we present a local discontinuous Galerkin （LDG） method for the AllenCahn equation. We prove the energy stability, analyze the optimal convergence rate of k ＋ 1 in L2 norm and present the （2k＋1）-th order negative-norm estimate of the semi- discrete LDG method for the Allen-Cahn equation with smooth solution. To relax the severe time step restriction of explicit time marching methods, we construct a first order semi-implicit scheme based on the convex splitting principle of the discrete Allen-Cahn energy and prove the corresponding unconditional energy stability. To achieve high order temporal accuracy, we employ the semi-implicit spectral deferred correction （SDC） method. Combining with the unconditionally stable convex splitting scheme, the SDC method can be high order accurate and stable in our numerical tests. To enhance the efficiency of the proposed methods, the multigrid solver is adapted to solve the resulting nonlinear algebraic systems. Numerical studies are presented to confirm that we can achieve optimal accuracy of （O（hk＋1） in L2 norm and improve the LDG solution from （O（hk＋1） to （O（h2k＋1） with the accuracy enhancement post-processing technique.