Uniformly Convergent Elements For Darcy-Stokes Problem And Discrete De Rham Complex

Differential equations with a small parameter ε multiplying with the highest order derivative terms are said to be singularly perturbed and normally boundary layers occur in their solutions. Singularly perturbed differential equations are usually arising from fields such as fluid mechanics, elasticity, acoustics, optics, optimal control, etc. For examples, the Darcy-Stokes equations, the Navier-Stokes equations of fluid flow at high Reynolds number,the equations governing flow in Porous media, and mathematical models of liquid crystal materials and of chemical reactions, etc. can reflect the physical character of the equations or can be introduced artificially. The perturbed parameters of such problems can reflect the physical character of the equations or can be introduced artificially, the solutions of these problems undergo rapid changes with in very thin subdomains, the true solution is important but indeterminacy.The singular perturbation solution of the problem usually associated with perturbed parameters ε, from this angle, we construct uniformly convergent finite elements, which have nothing with the parameter ε. This thesis mainly discuss singular perturbation prob-lem of Darcy-Stokes equations. We construct the finite element spaces that is uniformly convergence, and error estimates and the corresponding discrete de Rham complexes are given. The main body is composed of three parts.In the first part, we consider a singular perturbation problem which describes2D flow. An H(div)-conforming rectangular element, DSR14, is proposed and analyzed first. This element has14degrees of freedom for velocity and is proved to be uniformly convergent with respect to perturbation constant. We then simplify this element to get another H(div)-conforming rectangular element, DSR12, which has12degrees of free-dom for velocity. The uniform convergence is also obtained for this element. Finally, we construct a discrete de Rham complex corresponding to DSR12element. In the second part, we construct a cubic element named DSC33for the Darcy-Stokes problem of three dimension. The finite element space Vh for velocity is H(div)-conforming, i.e., the normal component of a function in Vh is continuous across the ele-ment boundaries, meanwhile the tangential component of a function in Vh is average con-tinuous across the element boundaries, hence Vh is H1-average conforming. We prove that this element is uniform convergent respect with the perturbation constant ε for the Darcy-Stokes problem. On the other hand we construct discrete de Rham complex corre-sponding to DSC33element. The finite element spaces in the discrete de Rham complex can be used in some singular perturbation problems.In the last part, we construct a tetrahedral element named DST20for the Darcy-Stokes problem of three dimension, this element simplify the degrees of velocity in [93]. The finite element spaces Vh for velocity are H(div)-conforming, i.e., the normal com-ponent of a function in Vh is continuous across the element boundaries, meanwhile the tangential component of a function in Vh is average continuous across the element bound-aries, hence Vh is H1-average conforming. We prove that this element is uniform conver-gent respect with the perturbation constant ε for the Darcy-Stokes problem. On the other hand we give a discrete de Rham complex corresponding to DST20element.