Search

In this paper a two grid algorithm for quasilinear elliptic problem based on virtual element method (VEM) discretization is proposed. With this new algorithm the solution of a quasilinear elliptic problem on a fine grid is reduced to the solution of a quasilinear elliptic problem on a much coarser grid, and the solution of a linear system on the fine grid. A priori error estimate in H1 norm is derived. Numerical experiments are carried out to illustrate the theoretical findings.

Photodynamic therapy (PDT) is highly effective in treating tumors located near body surface, offering strong tumor suppression and low damage to normal tissue nearby. PDT is also effective for treating a number of other conditions. PDT not only provide a precise and selective method for the treatment of various diseases by itself, it can also be used in combination with other traditional therapies. Because PDT uses light as the unique targeting mechanism, it has simpler and more direct targeting capability than traditional therapies. The core material of a PDT system is the photosensitizer which converts light energy to therapeutic factors/substances. Different photosensitizers have their distinct characteristics, leading to different advantages and disadvantages. These could be enhanced or compensated by using proper PDT system. Therefore, the selected type of photosensitizer would heavily influence the overall design of a PDT system. In this article, we evaluated major types of inorganic and organic PDT photosensitizers, and discussed future research directions in the field.

In this paper we investigate the finite element approximation of time fractional optimal control problem with integral state constraint. A space-time finite element scheme for the control problem is developed with piecewise constant time discretization and piecewise linear spatial discretization for the state equation. A priori error estimate for the space-time discrete scheme is derived. Projected gradient algorithm is used to solve the discrete optimal control problem. Numerical experiments are carried out to illustrate the theoretical findings.

In this paper spectral Galerkin approximation of optimal control problem governed by fractional advection diffusion reaction equation with integral state constraint is investigated. First order optimal condition of the control problem is discussed. Weighted Jacobi polynomials are used to approximate the state and adjoint state. A priori error estimates for control, state, adjoint state and Lagrangian multiplier are derived. Numerical experiment is carried out to illustrate the theoretical findings.

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.

In this paper we investigate a semi-discrete H1 -Galerkin mixed finite element approximation of one kind of nolocal second order nonlinear hyperbolic equation, which is often used to describe vibration of an elastic string. A priori error estimates for the semi-discrete scheme are derived. A fully discrete scheme is constructed and one numerical example is given to verify the theoretical findings.

In this paper, we investigate a mixed discontinuous Galerkin approximation of time dependent convection diffusion optimal control problem with control constraints based on the combination of a mixed finite element method for the elliptic part and a discontinuous Galerkin method for the hyperbolic part of the state equation. The control variable is approximated by variational discretization approach. A priori error estimates of the state, adjoint state, and control are derived for both semidiscrete scheme and fully discrete scheme. Numerical example is given to show the effectiveness of the numerical scheme.

We investigate an H1-Galerkin expanded mixed finite element approximation of nonlinear second-order hyperbolic equations, which model a wide variety of phenomena that involve wave motion or convective transport process. This method possesses some features such as approximating the unknown scalar, its gradient, and the flux function simultaneously, the finite element space being free of LBB condition, and avoiding the difficulties arising from calculating the inverse of coefficient tensor. The existence and uniqueness of the numerical solution are discussed. Optimal-order error estimates for this method are proved without introducing curl operator. A numerical example is also given to illustrate the theoretical findings.

In this paper, we study an adaptive finite element approximation of optimal control problems with integral fractional Laplacian and pointwise control constraints. The state variable is approximated by piecewise linear polynomials, and the control variable is implicitly discretized. Upper and lower bounds of a posteriori error estimates for finite element approximation of the optimal control problem are derived. An h-adaptive algorithm driven by the a posterior error estimator is presented with Dörfler’s marking criterion. We prove that the adaptive algorithm yields a sequence of approximations that converge at the optimal algebraic rate. Numerical examples are given to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

We present an H¹-Galerkin mixed finite element method for the solution of a nonlinear parabolic pressure equation, which arises in the mathematical models for describing a compressible fluid flow process in subsurface porous media. The method possesses the advantages of mixed finite element methods while avoiding directly inverting the permeability tensor, which is important especially in a low permeability zone. We conducted theoretical analysis to study the existence and uniqueness of the numerical solutions of the scheme and prove an optimal-order error estimate for the method. Numerical experiments are performed to justify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper we analyze the Local Discontinuous Galerkin (LDG) method for the constrained optimal control problem governed by the unsteady convection diffusion equations. A priori error estimates are obtained for both the state, the adjoint state and the control. For the discretization of the control we discuss two different approaches which have been used for elliptic optimal control problem. [ABSTRACT FROM AUTHOR]