We define a multilevel finite element discretization for a coupled stationary reaction-diffusion system in which each component can be defined on a separate grid. We prove convergence of the scheme and propose residual a-posteriori estimators for the error in the natural energy norm for the system. The estimators are robust in the coefficients of the system. We prove upper and lower bounds and illustrate the theory with numerical experiments.

We develop a multiscale discontinuous Galerkin (DG) method for solving a class of second order elliptic problems with rough coefficients. The main ingredient of this method is to use a non-polynomial multiscale approximation space in the DG method to capture the multiscale solutions using coarse meshes without resolving the fine scale structure of the solution. Theoretical proofs and numerical examples are presented in both one and two dimensions. For one-dimensional problems, optimal error estimates and numerical examples are shown for arbitrary order approximations. For two-dimensional problems, numerical results are presented by the high order multiscale DG method, but the error estimate is proven only for the second order method.

In this paper, we investigate a posteriori error estimates of the $hp$-finite element method for a distributed convex optimal control problem governed by the elliptic partial differential equations. A family of weighted a posteriori error estimators of residual type are formulated. Both reliability and efficiency of the estimators are analyzed.

In this paper, we present a new pressure-correction projection scheme for solving the time-dependent Navier-Stokes equations, which is based on the Crank-Nicolson extrapolation method in the time discretization. Error estimates for the velocity and the pressure of semidiscretized scheme are derived by interpreting the projection scheme as second-order time discretization of a perturbed system which approximates the incompressible Navier-Stokes equations.

We conduct a systematic comparison of a spectral collocation method with some symplectic methods in solving Hamiltonian dynamical systems. Our main emphasis is on non-linear problems. Numerical evidence has demonstrated that the proposed spectral collocation method preserves both energy and symplectic structure up to the machine error in each time (large) step, and therefore has a better long time behavior.

In this paper, we extend our previous work on the two-dimensional immersed boundary method for interfacial flows with insoluble surfactant to the case of three-dimensional axisymmetric interfacial flows. Although the key components of the scheme are similar in spirit to the two-dimensional case, there are two differences introduced in the present work. Firstly, the governing equations are written in an immersed boundary formulation using the axisymmetric cylindrical coordinates. Secondly, we introduce an artificial tangential velocity to the Lagrangian markers so that the uniform distribution of markers along the interface can be achieved and a modified surfactant concentration equation is derived as well. The numerical scheme still preserves the total mass of surfactant along the interface. Numerical convergence of the present scheme has been checked, and several tests for a drop in extensional flows have been investigated in detail.

We study extensions of the energy and helicity preserving scheme for the 3D Navier-Stokes equations, developed in [23], to a more general class of problems. The scheme is studied together with stabilizations of grad-div type in order to mitigate the effect of the Bernoulli pressure error on the velocity error. We prove stability, convergence, discuss conservation properties, and present numerical experiments that demonstrate the advantages of the scheme.

Recently, it is shown that graph cuts algorithms can be used to solve some variational image restoration problems, especially connected with noise removal and segmentation. For very large size images, the usage for memory and computation increases dramatically. We propose a domain decomposition method with graph cuts algorithms. We show that the new approach costs effective both for memory and computation. Experiments with large size 2D and 3D data are supplied to show the efficiency of the algorithms.

In this paper, we propose a domain decomposition method with Lagrange multipliers for three-dimensional linear elasticity, based on geometrically non-conforming subdomain partitions. Some appropriate multiplier spaces are presented to deal with the geometrically non-conforming partitions, resulting in a discrete saddle-point system. An augmented technique is introduced, such that the resulting new saddle-point system can be solved by the existing iterative methods. Two simple inexact preconditioners are constructed for the saddle-point system, one for the displacement variable, and the other for the Schur complement associated with the multiplier variable. It is shown that the global preconditioned system has a nearly optimal condition number, which is independent of the large variations of the material parameters across the local interfaces.

We consider the optimal $\mathcal{H}_2$ model reduction for large scale multi-input multi-output systems via tangential interpolation. Specifically, we prove that for general multi-input multi-output systems, the tangential interpolation-based optimality conditions and the gramian-based optimality conditions are equivalent. Based on the tangential interpolation, a fast algorithm is proposed for the optimal $\mathcal{H}_2$ model reduction. Numerical examples are presented to demonstrate the approximation accuracy and computational efficiency of the proposed algorithm.