Multilevel augmentation methods for solving the Burgers' equation

In this article, multilevel augmentation method (MAM) for solving the Burgers' equation is developed. The Crank–Nicolson–Galerkin scheme of the Burgers' equation results in nonlinear algebraic systems at each time step, the computational cost for solving these nonlinear systems is huge. The MAM allows us to solve the nonlinear system at a fixed initial lower level and then compensate the error by solving a linear system at the higher level. We prove that the method has the same optimal convergence order as the projection method, while reducing the computational complexity greatly. Finally, numerical experiments are presented to confirm the theoretical analysis and illustrate the efficiency of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1665–1691, 2015     

A Fourier pseudospectral method for a generalized improved Boussinesq equation

In this article, we propose a Fourier pseudospectral method for solving the generalized improved Boussinesq equation. We prove the convergence of the semi‐discrete scheme in the energy space. For various power nonlinearities, we consider three test problems concerning the propagation of a single solitary wave, the interaction of two solitary waves and a solution that blows up in finite time. We compare our numerical results with those given in the literature in terms of numerical accuracy. The numerical comparisons show that the Fourier pseudospectral method provides highly accurate results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 995–1008, 2015     

An operator splitting approach for the interaction between a fluid and a multilayered poroelastic structure

We develop a loosely coupled fluid‐structure interaction finite element solver based on the Lie operator splitting scheme. The scheme is applied to the interaction between an incompressible, viscous, Newtonian fluid, and a multilayered structure, which consists of a thin elastic layer and a thick poroelastic material. The thin layer is modeled using the linearly elastic Koiter membrane model, while the thick poroelastic layer is modeled as a Biot system. We prove a conditional stability of the scheme and derive error estimates. Theoretical results are supported with numerical examples. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1054–1100, 2015     

The Euler-Galerkin finite element method for nonlocal diffusion problems with a p-Laplace-type operator

This paper is devoted to the study of the finite element method for a class of non-linear nonlocal diffusion problems associated with p-Laplace-type operator. Using the Euler–Galerkin finite element method, the convergence and a priori error estimates for the semi-discrete as well as fully-discrete formulations are established.     

Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids

In this paper, we study the convergence of a finite difference scheme on nonuniform grids for the solution of second-order elliptic equations with mixed derivatives and variable coefficients in polygonal domains subjected to Dirichlet boundary conditions. We show that the scheme is equivalent to a fully discrete linear finite element approximation with quadrature. It exhibits the phenomenon of supraconvergence, more precisely, for s ∈ [1,2] order O(h ^s )-convergence of the finite difference solution, and its gradient is shown if the exact solution is in the Sobolev space H ^1+s (Ω). In the case of an equation with mixed derivatives in a domain containing oblique boundary sections, the convergence order is reduced to O(h ^3/2?ε) with ε > 0 if u ∈ H ³(Ω). The second-order accuracy of the finite difference gradient is in the finite element context nothing else than the supercloseness of the gradient. For s ∈ {1,2}, the given error estimates are strictly local.     

A Second-Order High-Resolution Scheme for a Juvenile-Adult Model of Amphibians

In this article we consider a juvenile-adult population model of amphibians in which juveniles are structured by age and adults are structured by size. We develop a second-order explicit high-resolution scheme to approximate the solution of the model. Convergence of the finite difference approximation to the unique weak solution with bounded total variation is proved. Numerical examples demonstrate the high-resolution property and the achievement of the designed accuracy for the scheme. The scheme is then applied to understand the dynamics of an urban amphibian population.     

Algebraic Multiplicities Arising from Static Feedback Control Systems of Parabolic Type

In stabilization studies of linear parabolic control systems, a successful approach is a scheme employing dynamic compensators in the feedback loop. An essential reason is the fact that both sensors and actuators cannot be designed freely, especially in the case of boundary observation/boundary feedback. Most fundamental in this scheme is a simple stabilization result under the static feedback control scheme. In this scheme, little attention has been paid to how to assign new eigenvalues of the feedback system. In this article, we show a new feature of pole assignment that shows some choices of new eigenvalues cause a deterioration of the stability property. An algebraic growth rate is added to the feedback system in such a choice.     

Pseudo-moderate deviations in the euler method for real diffusion process

We consider the problem of the non-sequential detection of a change in the drift coefficient of a stochastic differential equation, when a misspecified model is used. We formulate the generalized likelihood ratio (GLR) test for this problem, and we study the behaviour of the associated error probabilities (false alarm and nodetection) in the small noise asymptotics. We obtain the following robustness result: even though a wrong model is used, the error probabilities go to zero with exponential rate, and the maximum likelihood estimator (MLE) of the change time is consistent, provided the change to be detected is larger (in some sense) than the misspecification error. We give also computable bounds for selecting the threshold of the test so as to achieve these exponential rates.