Pursuing our work in [18], [17], [20], [5], we consider in this article the two-dimensional thermohydraulics equations. We discretize these equations in time using the implicit Euler scheme and we prove that the global attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.

In this paper, the mathematical formulation for a quadratic optimal control problem governed by a linear quasi-parabolic integro-differential equation is studied, the optimality conditions are derived, and then the a priori error estimate for its finite element approximation is given. Furthermore, some numerical tests are performed to verify the theoretical results.

This paper is concerned with residual type a posteriori error estimators for finite element methods for the Stokes equations. In particular, the authors established a unified approach for deriving and analyzing a posteriori error estimators for velocity-pressure based finite element formulations for the Stokes equations. A general a posteriori error estimator was presented with a unified mathematical analysis for the general finite element formulation that covers conforming, non-conforming, and discontinuous Galerkin methods as examples. The key behind the mathematical analysis is the use of a lifting operator from discontinuous finite element spaces to continuous ones for which all the terms involving jumps at interior edges disappear.

In this paper we use the Arakawa Jacobian method [1] and the fourth-order essentially non-oscillatory (ENO-4) scheme of Osher and Shu [15] to solve the equatorial beta-plane barotropic equations. The Arakawa Jacobian scheme is a second order centred finite differences scheme that conserves energy and enstrophy. The fourth-order essentially non-oscillatory scheme is designed for Hamilton-Jacobi equations and traditionally used to track sharp fronts. We are interested in the performance of these two methods on the baratropic equations and determine whether they are adequate for studying the barotropic instability. The two methods are tested and compared on two typical exact solutions, a smooth Rossby wave-packet and a discontinuous shear, on the long-climate scale of 100 days. The numerical results indicate that the Arakawa Jacobian method conserves energy and enstrophy nearly exactly, as expected, captures the phase speed the Rossby wave, and achieves an overall second order accuracy, in both cases. The same properties are preserved by the ENO-4 scheme but the fourth order accuracy is observed only for the smooth Rossby wave solution while in the case of the discontinuous shear, it yields an overall third order accuracy, even in the smooth regions, away from the discontinuity.

Linear elements are least expensive finite elements for simultaneously recovering the source location and intensity in a general convection-diffusion process. However, the derivatives of the least-squares objective functional with Tikhonov regularizations are not well-defined when linear finite elements are used. In this work we provide a systematic formulation of the numerical inversion using linear finite elements and propose some effective techniques to overcome the undefinedness that may occur in inversion process. We show that linear finite elements can be made very robust and efficient in simultaneously recovering the source location and intensity. Numerical results are presented to validate the robustness and effectiveness of the proposed algorithm.

We present a high order parameter-robust finite difference method for a linear system of ($M\geq2$) coupled singularly perturbed reaction-diffusion two point boundary value problems. The problem is discretized using a suitable combination of the fourth order compact difference scheme and the central difference scheme on a generalized Shishkin mesh. A high order decomposition of the exact solution into its regular and singular parts is constructed. The error analysis is given and the method is proved to have almost fourth order parameter robust convergence, in the maximum norm. Numerical experiments are conducted to demonstrate the theoretical results.

This paper is concerned with the time-step condition of commonly-used linearized semi-implicit schemes for nonlinear parabolic PDEs with Galerkin finite element approximations. In particular, we study the time-dependent nonlinear Joule heating equations. We present optimal error estimates of the semi-implicit Euler scheme in both the $L^2$ norm and the $H^1$ norm without any time-step restriction. Theoretical analysis is based on a new splitting of error function and precise analysis of a corresponding time-discrete system. The method used in this paper is applicable for more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations for which previous works often require certain restriction on the time-step size $\tau$.

A hybrid stress finite volume method is proposed for linear elasticity equations. In this new method, a finite volume formulation is used for the equilibrium equation, and a hybrid stress quadrilateral finite element discretization, with continuous piecewise isoparametric bilinear displacement interpolation and two types of stress approximation modes, is used for the constitutive equation. The method is shown to be free from Poisson-locking and of first order convergence. Numerical experiments confirm the theoretical results.

In this paper, we develop a mixed Fourier-generalized Jacobi rational spectral method for two-dimensional exterior problems. Some basic results on the mixed Fourier-generalized Jacobi rational orthogonal approximations are established. Two model problems are considered. The convergence for the linear problem is proved. Numerical results demonstrate its spectral accuracy and efficiency.

The minimization of the energy functional having states constrained to semi-linear parabolic PDEs is considered. The controls act on the boundary and are of Robin type. The discrete schemes under consideration are discontinuous in time but conforming in space. Stability estimates are presented at the energy norm and at arbitrary times for the state, and adjoint variables. The estimates are derived under minimal regularity assumptions and are applicable for higher order elements. Using these estimates and an appropriate compactness argument (see Walkington [49, Theorem 3.1]) for discontinuous Galerkin schemes, convergence of the discrete solution to the continuous solution is established. In addition, a discrete optimality system is derived and convergence of the corresponding discrete solutions is also demonstrated.

In this paper, finite volume element method is applied to solve the distributed optimal control problems governed by an elliptic equation. We use the method of variational discretization concept to approximate the problems. The optimal order error estimates in $L^2$ and $L^∞$-norm are derived for the state, costate and control variables. The optimal $H^1$ and $W^{1,∞}$-norm error estimates for the state and costate variables are also obtained. Numerical experiments are presented to test the theoretical results.

We consider an approximation scheme for systems of linear delay-differential equations of neutral type. The finite dimensional approximating systems are constructed with basis functions defined using linear splines, extending to neutral equations a scheme which had previously been defined only for retarded equations. A Trotter-Kato semigroup convergence result is proved, and numerical results are given to illustrate the qualitative behavior of the scheme.

In this paper we study the finite element approximation for stochastic Navier-Stokes equations including a turbulent part. The discretization for space is derived by finite element method, and we use the backward Euler scheme in time discretization. We apply the generalized $L_2$-projection operator to approximate the noise term. Under suitable assumptions, strong convergence error estimations with respect to the fully discrete scheme are well proved.

A system of linear equations $Ax = b$, in $n$ unknowns and $m$ equations which has a nonnegative solution is considered. Among all its solutions, the one which has the least norm is sought when $\mathbb{R}^n$ is equipped with a strictly convex norm. We present a globally convergent, iterative algorithm for computing this solution. This algorithm takes into account the special structure of the problem. Each iteration cycle of the algorithm involves the solution of a similar quadratic problem with a modified objective function. Duality conditions for optimality are studied. Feasibility and global convergence of the algorithm are proved. As a special case we implemented and tested the algorithm for the $\ell^p$-norm, where $1 < p < ∞$. Numerical results are included.