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This paper aims at designing an observer-based feedback law which locally stabilizes the solution to the two dimensional Navier–Stokes equations with mixed boundary conditions. We consider a finite number of controls acting on a portion of the boundary through Robin boundary conditions and construct a linear Luenberger observer based on the point observations of the linearized Navier–Stokes equations. The sensor location for the point observations is determined by the response of feedback functional gains. We prove that the nonlinear system coupled with the observer through the feedback law is locally exponentially stable. Numerical experiments based on a Taylor–Hood finite element method are presented to illustrate the design for different Reynolds numbers. [ABSTRACT FROM AUTHOR]

In this paper, we discuss finite element methods for the incompressible Stokes problem and the nearly incompressible linear elasticity problem. Specifically, we present a finite element pair for the incompressible Stokes problem, which satisfies the discrete inf–sup condition and the discrete Korn’s inequality, and moreover, which is element-wise conservative. The pair provides a locking-free method for the nearly incompressible linear elasticity problem without reduced integration. [ABSTRACT FROM AUTHOR]

In this paper, the decoupled stabilized finite element methods are developed for the Boussinesq equations with temperature dependent coefficients. In our numerical schemes, the original problem is decoupled into the linearized Navier-Stokes equations and a parabolic problem, the low order mixed finite element pairs are used to approximate the spatial spaces and the backward Euler scheme is adopted to treat the time terms. Furthermore, the linear terms are dealt with the implicit scheme while the nonlinear terms are treated by the semi-implicit scheme, then, a lot of storage and computational cost are saved. The advantages for our numerical schemes are parameter-free, unconditionally stable, and constant matrix at each time level. Finally, some numerical tests are presented to verify the theoretical results of the developed numerical methods, and show that our schemes not only keep good accuracy but also save a lot of computational times. [ABSTRACT FROM AUTHOR]

Mixed finite elements play a central role in many important CFD applications involving Stokes solvers and alike. A natural mixed finite element for the Stokes equations is the Q k - Q k − 1 element on rectangular grids, by which the velocity is approximated by continuous polynomials of separated degree k and the pressure is approximated by discontinuous polynomials of separated degree k − 1 . Such an element is, however, not stable. We propose in this paper three modified Q k - Q k − 1 elements with certain element-wise divergence-free property of velocity, where the pressure space is slightly restricted to subspaces in C − 1 - Q k − 1 , yet the optimal order of approximation is still retained. The stability and approximation analysis for the new elements are presented. Comprehensive numerical experiments are also conducted to confirm the theoretical analysis and to reveal the super-convergence for some of these new elements. [ABSTRACT FROM AUTHOR]

This paper characterizes one penalty finite element method for the incompressible MHD equations. The method is an interesting combination of the classic iterative schemes (Stokes, Newton and Oseen iterations) with two different finite element pairs P 1 b – P 1 – P 1 b and P 1 – P 0 – P 1 . Moreover, the rigorous analysis of stability and error estimate for the proposed methods are given. Finally, the applicability and effectiveness of the presented schemes are illustrated in several numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we construct new nonconforming finite elements on the meshes consisting of arbitrary convex quadrilaterals, especially for the quadratic and cubic cases. For each case, we first define a quadrilateral element that adopts edge moments as the degrees of freedom (DoFs), and then enforce a linear constraint on this element. We have, for the quadratic case, eight degrees of freedom per element and, for the cubic case, eleven DoFs per element, respectively. The dimensions and the bases of different types for the global finite element spaces are provided. We consider the approximations of two-dimensional second order elliptic problems for both of these elements. Error estimates with optimal convergence order in both broken H 1 norm and L 2 norm are given. Moreover, we consider the discretization of the Stokes equations adopting our quadratic element to approximate each component of the velocity, along with piecewise discontinuous P 1 element for the pressure. This mixed scheme is stable and optimal error estimates both for the velocity and the pressure are also achieved. Numerical examples verify our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper we analyze time marching schemes for the wave equation in mixed form. The problem is discretized in space using stabilized finite elements. On the one hand, stability and convergence analyses of the fully discrete numerical schemes are presented using different time integration schemes and appropriate functional settings. On the other hand, we use Fourier techniques (also known as von Neumann analysis) in order to analyze stability, dispersion and dissipation. Numerical convergence tests are presented for various time integration schemes, polynomial interpolations (for the spatial discretization), stabilization methods, and variational forms. To analyze the behavior of the different schemes considered, a 1D wave propagation problem is solved. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we develop numerical approximations of the wave equation in mixed form supplemented with non-reflecting boundary conditions (NRBCs) of Sommerfeld-type on artificial boundaries for truncated domains. We consider three different variational forms for this problem, depending on the functional space for the solution, in particular, in what refers to the regularity required on artificial boundaries. Then, stabilized finite element methods that can mimic these three functional settings are described. Stability and convergence analyses of these stabilized formulations including the NRBC are presented. Additionally, numerical convergence test are evaluated for various polynomial interpolations, stabilization methods and variational forms. Finally, several benchmark problems are solved to determine the accuracy of these methods in 2D and 3D. [Copyright &y& Elsevier]