We study competitive economy equilibrium computation. We show that, for the first time, the equilibrium sets of the following two markets: 1. A mixed Fisher and Arrow-Debreu market with homogeneous and log-concave utility functions; 2. The Fisher and Arrow-Debreu markets with several classes of concave non-homogeneous utility functions; are convex or log-convex. Furthermore, an equilibrium can be computed as convex optimization by an interior-point algorithm in polynomial time.

In an earlier paper, we proved the existence of solutions to the Skorohod problem with oblique reflection in time-dependent domains and, subsequently, applied this result to the problem of constructing solutions, in time-dependent domains, to stochastic differential equations with oblique reflection. In this paper we use these results to construct weak approximations of solutions to stochastic differential equations with oblique reflection, in time-dependent domains in $\mathbb{R}^{d}$, by means of a projected Euler scheme. We prove that the constructed method has, as is the case for normal reflection and time-independent domains, an order of convergence equal to $1/2$ and we evaluate the method empirically by means of two numerical examples. Furthermore, using a well-known extension of the Feynman-Kac formula, to stochastic differential equations with reflection, our method gives, in addition, a Monte Carlo method for solving second order parabolic partial differential equations with Robin boundary conditions in time-dependent domains.

In this paper, we present further development of the local discontinuous Galerkin (LDG) method designed in [21] and a new dissipative discontinuous Galerkin (DG) method for the Hunter-Saxton equation. The numerical fluxes for the LDG and DG methods in this paper are based on the upwinding principle. The resulting schemes provide additional energy dissipation and better control of numerical oscillations near derivative singularities. Stability and convergence of the schemes are proved theoretically, and numerical simulation results are provided to compare with the scheme in [21].

In this paper we study the convergence of adaptive finite element methods for the general non-affine equivalent quadrilateral and hexahedral elements on 1-irregular meshes with hanging nodes. Based on several basic ingredients, such as quasi-orthogonality, estimator reduction and Döfler marking strategy, convergence of the adaptive finite element methods for the general second-order elliptic partial equations is proved. Our analysis is effective for all conforming $Q_m$ elements which covers both the two- and three-dimensional cases in a unified fashion.

In this paper, we study adaptive finite element discretisation schemes for a class of parameter estimation problem. We propose to use adaptive multi-meshes in developing efficient algorithms for the estimation problem. We derive equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive multi-mesh finite element scheme. The error estimators are then implemented and tested with promising numerical results.

In this paper, we extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by Lions et al. [12] and solve a nonlinear evolutionary parabolic partial differential equation. The fine solver is based on the finite element method or spectral element method in space and a semi-implicit Runge-Kutta scheme in time. The coarse solver is based on a semi-implicit scheme in time and the reduced basis approximation in space. Offline-online procedures are developed, and it is proved that the computational complexity of the on-line stage depends only on the dimension of the reduced basis space (typically small). Parareal in time algorithms based on a multi-grids finite element method and a multi-degrees finite element method are also presented. Some numerical results are reported.

In this paper, we consider the time dependent Maxwell's equations when dispersive media are involved. The Crank-Nicolson mixed finite element methods are developed for three most popular dispersive medium models: the isotropic cold plasma, the one-pole Debye medium and the two-pole Lorentz medium. Optimal error estimates are proved for all three models solved by the Raviart-Thomas-Nédélec spaces. Extensions to multiple pole dispersive media are presented also.

A numerical method based on finite difference method with variable mesh is given for self-adjoint singularly perturbed two-point boundary value problems. To obtain parameter-uniform convergence, a variable mesh is constructed, which is dense in the boundary layer region and coarse in the outer region. The uniform convergence analysis of the method is discussed. The original problem is reduced to its normal form and the reduced problem is solved by finite difference method taking variable mesh. To support the efficiency of the method, several numerical examples have been considered.