We design several implicit asymptotic-preserving schemes for the linear semiconductor Boltzmann equation with a diffusive scaling, which lead asymptotically to the implicit discretizations of the drift-diffusion equation. The constructions are based on a stiff relaxation step and a stiff convection step obtained by splitting the system equal to the model equation. The one space dimensional schemes are given with the uniform grids and the staggered grids, respectively. The uniform grids are considered only in two space dimension. The relaxation step is evolved with the BGK-penalty method of Filbet and Jin [F. Filbet and S. Jin, J. Comp. Phys. 229(20), 7625-7648, 2010], which avoids inverting the complicated nonlocal anisotropic collision operator. The convection step is performed with a suitable implicit approximation to the convection term, which gives a banded matrix easy to invert. The von-Neumman analysis for the Goldstein-Taylor model show that the one space dimensional schemes are unconditionally stable. The heuristic discussions suggest that all the proposed schemes have the correct discrete drift-diffusion limit. The numerical results verify that all the schemes are asymptotic-preserving. As far as we know, they are the first class of asymptotic-preserving schemes ever introduced for the Boltzmann equation with a diffusive scaling that lead to an implicit discretization of the diffusion limit, thus significantly relax to stability condition.

A time-dependent convection-diffusion problem is discretized by the Galerkin finite element method in space with bilinear elements on a general layer adapted mesh and in time by discontinuous Galerkin method. We present optimal error estimates. The estimates hold true for consistent stabilization too.

This work is concerned with the numerical approximation of the unsteady Stokes flow of a viscous incompressible fluid driven by a threshold slip boundary condition of friction type. The continuous problem is formulated as variational inequality, which is next discretized in time based on backward Euler's scheme. We prove existence and uniqueness of the solution of the time discrete problem by means of a regularization approach. Finally, we derive error estimates that justify the convergence property of the discretization proposed.

In this paper we hope to draw attention to a particularly simple and extremely flexible design strategy for solving a wide class of "set-point" regulation problems for nonlinear parabolic boundary control systems. By this we mean that the signals to be tracked and disturbances to be rejected are time independent. The theoretical underpinnings of our approach is the well known regulator equations from the geometric theory of regulation applicable in the neighborhood of an equilibrium. The most important point of this work is the wide applicability of the design methodology. In the examples we have employed unbounded sensing and actuation but the method works equally well for bounded input and output operators and even finite dimensional nonlinear control systems. Our examples include: multi-input multi-output regulation for a boundary controlled viscous Burgers' equation; control of a Navier-Stokes flow in two dimensional forked channel; control problem for a non-Isothermal Navier-Stokes flow in two dimensional box domain. Along the way we provide some discussion to demonstrate how the method can be altered to provide many alternative control mechanisms. In particular, in the last section we show how the method can be adapted to solve tracking and disturbance rejection for piecewise constant time dependent signals.

The problem of a stationary generalized convective flow modelling bioconvection is considered. The viscosity is assumed to be a function of the concentration of the micro-organisms. As a result, the PDE system describing the bioconvection model is quasilinear. The existence and uniqueness of the weak solution of the PDE system is obtained under minimum regularity assumption on the viscosity. Numerical approximations based on the finite element method are constructed and error estimates are obtained. Numerical experiments are conducted to demonstrate the accuracy of the numerical method as well as to simulate bioconvection pattern formations based on realistic model parameters.

In this work, we propose a surface fitting strategy based on a two-step model to remove noise from digital images. In the first step, we minimize the total variation energy functional of an image by using the projection gradient method in order to obtain the dual variable as the smoothed normal vector. In the second step, we try to find a surface as the recovered image to fit the smoothed normal vector. Based on the projection gradient method and the variable splitting method, we propose an efficient numerical method to solve this two-step model and also give the convergence analysis of the proposed method. Some numerical comparisons are given to validate the effectiveness of our proposed model.

Fast sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilize the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. They take advantage of the properties of hyperbolic partial differential equations (PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. In [16], the Gauss-Seidel idea and alternating sweeping strategy were adopted to the time-marching type fixed-point iterations to solve the static Hamilton-Jacobi equations, and numerical examples verified at least a 2 times acceleration of convergence even on relatively coarse grids. In this paper, we apply the same approach to solve steady state solution of hyperbolic conservation laws. We use numerical examples to verify that a 2 times acceleration of convergence is achieved. And the computational cost is exactly the same as the time-marching scheme at each iteration. Based on the Gauss-Seidel iterations, we explore the successive overrelaxation (SOR) approach to further improve the performance of our fixed-point sweeping methods.

This paper deals with the numerical solution of both linear and non-linear Schrödinger problems, which mathematically model many physical processes in a wide range of applications of interest. In particular, a comparison of different solvers and different approaches for these problems is developed throughout this work. Two finite difference schemes are analyzed: the classical Crank-Nicolson approach, and a high-order compact scheme. Solvers based on geometric multigrid, Fast Fourier Transform and Alternating Direction Implicit methods are compared. Finally, the efficiency of the considered solvers is tested for a linear Schrödinger problem, proving that the computational experiments are in good agreement with the theoretical predictions. In order to test the robustness of the MG solver two additional Schrödinger problems with a nonconstant potential and nonlinear right-hand side are solved by the MG solver, since the efficiency of this solver depends on such data.

We study relaxation approximations to solutions of a 2 × 2 triangular system of conservation laws. We show that smooth relaxation approximations exist for all time. A finite difference approximation of the relaxation system gives rise to a relaxation scheme of the Jin and Xin type. In both cases we show that a sequence of approximate solutions is produced where the limit is a weak solution of the triangular system. Compensated compactness is used to establish convergence.

In this paper we study the global convergence of the implicit residual-based a posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional linear hyperbolic problems. We apply a new optimal superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the $L^2$-norm under mesh refinement. The order of convergence is proved to be $k+2$, when $k$-degree piecewise polynomials with $k\geq1$ are used. As a consequence, we prove that the DG method combined with the a posteriori error estimation procedure yields both accurate error estimates and $\mathcal{O}(h^{k+2})$ superconvergent solutions. We perform numerical experiments to demonstrate that the rate of convergence is optimal. We further prove that the global effectivity indices in the $L^2$-norm converge to unity under mesh refinement. The order of convergence is proved to be 1. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be $k+3/2$ and $1/2$, respectively. Our proofs are valid for arbitrary regular meshes using $P^k$ polynomials with $k\geq1$ and for both the periodic boundary condition and the initial-boundary value problem. Several numerical simulations are performed to validate the theory.

A hybrid operator splitting method is developed for computations of two-dimensional transverse magnetic Maxwell equations in media with multiple random interfaces. By projecting the solutions into the random space using the polynomial chaos (PC) projection method, the deterministic and random parts of the solutions are solved separately.
There are two independent stages in the algorithm: the Yee scheme with domain decomposition implemented on a staggered grid for the deterministic part and the Monte Carlo sampling in the post-processing stage. These two stages of the algorithm are subject of computational studies. A parallel implementation is proposed for which the computational cost grows linearly with the number of random interfaces. Output statistics of Maxwell solutions are obtained including means, variance and time evolution of cumulative distribution functions (CDF). The computational results are presented for several configurations of domains with random interfaces.
The novelty of this article lies in using level set functions to characterize the random interfaces and, under reasonable assumptions on the random interfaces (see Figure 1), the dimensionality issue from the PC expansions is resolved (see Sections 1.1.2 and 1.2).

A new reduced model for the shallow tridimensional viscoplastic fluid is presented in this paper, allowing for the first time an arbitrarily topography. A new numerical approach is also proposed in order to catch efficiently the long-time behavior of the flow and the arrested state. In order to support varying and large time steps, a fully implicit and second order method (BDF2) is proposed. It is combined with an auto-adaptive mesh feature for catching accurately the evolution of front position. This approach was tested on two flows experiments and compared to experimental measurements. The first study shows the efficiency of this approach when the shallow flow conditions are fully satisfied while the second one points out the limitations of the reduced model when these conditions are not fulfilled.

In this work, we show the convergence of adaptive lowest-order FEM (AFEM) for an elliptic obstacle problem with non-homogeneous Dirichlet data, where the obstacle $\chi$ is restricted only by $\chi\in H^2(\Omega)$. The adaptive loop is steered by some residual based error estimator introduced in Braess, Carstensen & Hoppe (2007) that is extended to control oscillations of the Dirichlet data, as well. In the spirit of Cascon ET AL. (2008), we show that a weighted sum of energy error, estimator, and Dirichlet oscillations satisfies a contraction property up to certain vanishing energy contributions. This result extends the analysis of Braess, Carstensen & Hoppe (2007) and Page & Praetorius (2013) to the case of non-homogeneous Dirichlet data as well as certain non-affine obstacles and introduces some energy estimates to overcome the lack of nestedness of the discrete spaces.