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.

Use of Total Precipitable Water Classification of A Priori Error and Quality Control in Atmospheric Temperature and Water Vapor Sounding Retrieval

This study investigates the use of dynamic a priori error information according to atmospheric moistness and the use of quality controls in temperature and water vapor profile retrievals from hyperspectral infrared （IR） sounders. Temperature and water vapor profiles are retrieved from Atmospheric InfraRed Sounder （AIRS） radiance measurements by applying a physical iterative method using regression retrieval as the first guess. Based on the dependency of first-guess errors on the degree of atmospheric moistness, the a priori first-guess errors classified by total precipitable water （TPW） are applied in the AIRS physical retrieval procedure. Compared to the retrieval results from a fixed a priori error, boundary layer moisture retrievals appear to be improved via TPW classification of a priori first-guess errors. Six quality control （QC） tests, which check non-converged or bad retrievals, large residuals, high terrain and desert areas, and large temperature and moisture deviations from the first guess regression retrieval, are also applied in the AIRS physical retrievals. Significantly large errors are found for the retrievals rejected by these six QCs, and the retrieval errors are substantially reduced via QC over land, which suggest the usefulness and high impact of the QCs, especially over land. In conclusion, the use of dynamic a priori error information according to atmospheric moistness, and the use of appropriate QCs dealing with the geographical information and the deviation from the first-guess as well as the conventional inverse performance are suggested to improve temperature and moisture retrievals and their applications.

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 OF A COMBINED MIXED FINITE ELEMENT AND DISCONTINUOUS GALERKIN METHOD FOR COMPRESSIBLE MISCIBLE DISPLACEMENT WITH MOLECULAR DIFFUSION AND DISPERSION

A combined approximation for a kind of compressible miscible displacement problems including molecular diffusion and dispersion in porous media is studied. Mixed finite el- ement method is applied to the ftow equation, and the transport one is solved by the symmetric interior penalty discontinuous Galerkin method （SIPG）. To avoid the inconve- nience of the cut-off operator in [3, 21], some induction hypotheses different from the ones in [6] are used. Based on interpolation projection properties, a priori hp error estimates are obtained. Comparing with the existing error analysis that only deals with the diffusion case, the current work is more complicated and more significant.

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 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 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.

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.

A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation

A new higher-order accurate space-time discontinuous Galerkin(DG)method using the interior penalty flux and discontinuous basis functions,both in space and in time,is pre-sented and fully analyzed for the second-order scalar wave equation.Special attention is given to the definition of the numerical fluxes since they are crucial for the stability and accuracy of the space-time DG method.The theoretical analysis shows that the DG discre-tization is stable and converges in a DG-norm on general unstructured and locally refined meshes,including local refinement in time.The space-time interior penalty DG discre-tization does not have a CFL-type restriction for stability.Optimal order of accuracy is obtained in the DG-norm if the mesh size h and the time stepΔt satisfy h≅CΔt,with C a positive constant.The optimal order of accuracy of the space-time DG discretization in the DG-norm is confirmed by calculations on several model problems.These calculations also show that for pth-order tensor product basis functions the convergence rate in the L∞and L2-norms is order p+1 for polynomial orders p=1 and p=3 and order p for polynomial order p=2.

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.

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.

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.

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.

AN ERROR ANALYSIS METHOD SPP-BEAM AND A CONSTRUCTION GUIDELINE OF NONCONFORMING FINITE ELEMENTS FOR FOURTH ORDER ELLIPTIC PROBLEMS

Under two hypotheses of nonconforming finite elements of fourth order elliptic problems,we present a side-patchwise projection based error analysis method(SPP-BEAM for short).Such a method is able to avoid both the regularity condition of exact solutions in the classical error analysis method and the complicated bubble function technique in the recent medius error analysis method.In addition,it is universal enough to admit generalizations.Then,we propose a sufficient condition for these hypotheses by imposing a set of in some sense necessary degrees of freedom of the shape function spaces.As an application,we use the theory to design a P3 second order triangular H2 non-conforming element by enriching two P4 bubble functions and,another P4 second order triangular H2 nonconforming finite element,and a P3 second order tetrahedral H2 non-conforming element by enriching eight P4 bubble functions,adding some more degrees of freedom.

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.

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.

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.