Several Nonconforming Finite Elements Over Quadrilateral Meshes

The finite element method is an effective numerical method for solving differential equa-tions,which is widely used in the field of scientific computing and engineering.And the non-conforming finite element method can obtain stable numerical solutions when solving practical problems in solid mechanics and fluid mechanics,such as solving problems related to elastic mechanics,Darcy-Stokes and Stokes problems.When the finite element method is used to solve differential equations,a single finite element consists of element geometry,shape function space and degrees of freedom.The finite element method is based on the partition of the solution re-gion,where triangulation is widely used and mesh generation is relatively simple.However,for some special areas,if quadrilateral mesh is used,the number of elements will be reduced consid-erably compared to triangular mesh,and the number of elements connected to each vertex will also be significantly reduced.This thesis mainly constructs three nonconforming finite elements for the quadrilateral sin-gular perturbation problem,Darcy-Stokes problem and Stokes problem,and proves its conver-gence theoretically.Finally,through specific numerical experiments,the convergence of these three finite elements is verified.The details are as follows:(1)For the fourth-order elliptic singular perturbation problem,we present a new C0 non-conforming quadrilateral element.For each convex quadrilateral Q,the shape function space is the union of spline space S21(Q*)and a bubble space.We define the DOFs as the eight values at the vertices and midpoints on each edge,and the four mean values of integrals of normal deriva-tives over edges.Twelve local basis functions are defined on each quadrilateral element.By introducing a new reference domain Q and an affine map,we can express the local basis func-tions explicitly without solving linear systems locally.Besides,all the integrations can be done over the reference domain,which is more efficient since the Jacobian determinant is constant.More precisely,we compute out the local stiffness matrix for each quadrilateral element.Our C0 finite element method is convergent uniformly with respect to the perturbation parameter for the fourth-order singular perturbation problem.(2)For the Darcy-Stokes problem,a new mixed finite element method is constructed over arbitrary convex quadrilateral meshes.We focus on H(div)-conforming finite element methods based on the classical velocity-pressure formulation and utilize the spline method.The velocity space is based on a spline space over quadrilateral,and we adopt piecewise constant element for approximating the pressure.Since this element is an H(div)-element,we need low regular-ity of the right-hand-side terms of the Darcy-Stokes problem.The discrete de Rham complex is constructed after the convergence results.Moreover,the basis representation can be shown explicitly through a Piola transformation.(3)A theoretical framework is constructed on how to develop non C0 nonconforming high convergence elements for the fourth order elliptic problem.By the theoretical analysis,we con-struct a element for solving the biharmonic problem on quadrilateral meshes.Then we construct a new mixed finite element method on quadrilateral grids for the Stokes problem.The veloc-ity shape involved only consist of polynomials.And we adopt piecewise constant element to approximate the pressure.For the Stokes element,all the degrees of freedom are on the edges.Note that such degree of freedom selection leads to a comparatively low band width.The discrete velocity is second-order convergent in the discrete H1-seminorm,and through a post-processing the convergence order of the pressure solution can be improved to be second.