In this article, we develop the semi-discrete and fully discrete averaging local discontinuous Galerkin method to solve the well-known Schrödinger equation, in which space is discretized by the averaging local discontinuous Galerkin (ADG) method, and the time is discretized by Crank-Nicolson approach. Energy and mass conservative property of both schemes are proved. These schemes are shown to be unconditionally energy stable, and the error estimates are rigorously proved. Some numerical examples are performed to demonstrate the accuracy numerically.

We study whether a modified version of Tikhonov regularization can be used to identify several local sources from Dirichlet boundary data for a prototypical elliptic PDE. This paper extends the results presented in [5]. It turns out that the possibility of distinguishing between two, or more, sources depends on the smoothing properties of a second or fourth order PDE. Consequently, the geometry of the involved domain, as well as the position of the sources relative to the boundary of this domain, determines the identifiability. We also present a uniqueness result for the identification of a single local source. This result is derived in terms of an abstract operator framework and is therefore not only applicable to the model problem studied in this paper. Our schemes yield quadratic optimization problems and can thus be solved with standard software tools. In addition to a theoretical investigation, this paper also contains several numerical experiments.

A class of index-1 integro-differential algebraic equations modeling a hydraulic circuit that feed a combustion process is considered. The existence, uniqueness and regularity are analyzed in detail. Two kinds of collocation methods are employed to solve the equation numerically. For the first one, the derivative and algebraic components are approximated in globally continuous and discontinuous polynomial spaces, respectively; and for another one, both the derivative and algebraic components are solved in globally continuous piecewise polynomial spaces. The convergence, global and local superconvergence are described for these two classes of collocation methods. Some numerical experiments are given to illustrate the obtained theoretical results.

A new local discontinuous Galerkin (LDG) method for convection-diffusion equations on overlapping meshes with periodic boundary conditions was introduced in [14]. With the new method, the primary variable $u$ and the auxiliary variable $p = u_x$ are solved on different meshes. In this paper, we will extend the idea to convection-diffusion equations with non-periodic boundary conditions, i.e. Neumann and Dirichlet boundary conditions. The main difference is to adjust the boundary cells. Moreover, we study the stability and suboptimal error estimates. Finally, numerical experiments are given to verify the theoretical findings.

This paper designs a decoupled scheme for approximating nematic liquid crystal flow based on a fully discrete mixed finite element method, which allows different time steps for different physical fields. Besides, error estimates for velocity and macroscopic molecular orientation of the nematic liquid crystal flow are shown. Finally, numerical tests are provided to demonstrate efficiency of the scheme. It is found the presented scheme can save lots of computational time compared with common decoupled scheme.

Positron emission tomography (PET) is traditionally modeled as discrete systems. Such models may be viewed as piecewise constant approximations of the underlying continuous model for the physical processes and geometry of the PET imaging. Due to the low accuracy of piecewise constant approximations, discrete models introduce an irreducible modeling error which fundamentally limits the quality of reconstructed images. To address this bottleneck, we propose an integral equation model for the PET imaging based on the physical and geometrical considerations, which describes accurately the true coincidences. We show that the proposed integral equation model is equivalent to the existing idealized model in terms of line integrals which is accurate but not suitable for numerical approximation. The proposed model allows us to discretize it using higher accuracy approximation methods. In particular, we discretize the integral equation by using the collocation principle with piecewise linear polynomials. The discretization leads to new ill-conditioned discrete systems for the PET reconstruction, which are further regularized by a novel wavelet-based regularizer. The resulting non-smooth optimization problem is then solved by a preconditioned proximity fixed-point algorithm. Convergence of the algorithm is established for a range of parameters involved in the algorithm. The proposed integral equation model combined with the discretization, regularization, and optimization algorithm provides a new PET image reconstruction method. Numerical results reveal that the proposed model substantially outperforms the conventional discrete model in terms of the consistency to simulated projection data and reconstructed image quality. This indicates that the proposed integral equation model with appropriate discretization and regularizer can significantly reduce modeling errors and suppress noise, which leads to improved image quality and projection data estimation.

This paper addressed the fractional continuous-time programming problem. The necessary and sufficient optimality conditions under generalized concavity assumptions are established. Dual problem to the primal one and investigating duality relations between them are also addressed.