Some Researches On Nonconforming Finite Elements And Projective Invariants

Finite element method is an effective tool for solving differential equations and many oth-er problems in engineering.It is valuable to design robust finite elements over various meshes and to construct reliable finite element schemes for given problems.Moreover,since projective geometry has close relations with algebraic geometry and computational geometry,it is an inter-esting work to apply projective invariants to other geometric objects.This work is dedicated to the researches of nonconforming finite elements and projective invariants.This thesis studies design and applications of nonconforming finite elements.First of all,in order to deal with second order elliptic problems,new robust nonconforming finite elements are designed over an arbitrary convex quadrilateral mesh,especially for the quadratic and cubic cases.For each case,we first define a nonconforming finite element over an arbitrary convex quadrilateral,whose degrees of freedom include the moments over the four edges of this quadri-lateral,and then enforce a linear constraint involving these moments to obtain a new constrained finite element.In the quadratic and cubic cases,each element has 8 and 11 degrees of freedom,respectively.The dimensions of the global finite element spaces depend on the numbers of cells,vertices and edges in the mesh.Moreover,a group of convenient basis for each space is provided for practical computation.The optimal convergence rate for second order elliptic problems are proved by using our elements.Numerical examples verify our theoretical analysis.Second-ly,this work designs dififerent mixed finite element methods for Stokes problems for different cases.The first method works for two-dimensional Stokes problems over the meshes consist-ing of arbitrary convex quadrilaterals.We utilize the robust finite elements provided in the first part with piecewise discontinuous polynomial elements to form stable mixed schemes.For the quadratic case,the direct piecewise discontinuous Pi element is a proper candidate for the pres?sure.For the cubic case,bubble functions are needed for the velocity,Then three-dimensional Stokes problems are discussed.The vector-valued MSLK element is adopted for the velocity over cuboid meshes.Since it is instable if one approximates the pressure by the standard piece-wise discontinuous Pi element directly,we modify the pressure space by selecting piecewise macro Pi element,and then derive the stability assertion.In contrast with some known methods,our method has less degrees of freedom without any reduction of convergence order.The next contribution is to give a correction to the work of Lamichhane in[61],where the stability of theCR-Pi element over simplicial meshes is asserted.Indeed,the stability of this pair severe-ly relies on the given mesh.For the two-dimensional case,this work provides and proves the necessary and sufficient condition for a macroelement which leads to the stability of the CR-P1 element.Furthermore,a modified version of this pair is given to adapt more general meshes.The stability and convergence assertions in all the three works above can be verified by numerical tests.This work also discusses applications of projective invariants in algebraic geometry and computational geometry.Firstly,a projective invariant is discovered in higher dimensional pro-jective space,which is an extension of the characteristic number of algebraic curves proposed by Professor Zhongxuan Luo,and so its name remains unchanged,although this quantity no longer relies on an algebraic curve.Then we apply this quantity to algebraic hypersurfaces and spline spaces as follows.Utilizing the c,haracteristic number,we provide an intersection property of an algebraic hypersurface and a set of lines constituting a closed loop.Two different forms of gen-eralizations of Pascal's theorem in higher dimensional space are given through this property,the first one establishing the connectlion of hypersurfaces of distinct degrees,and the other concerned with the intersections of a hypersurface and a simplex.These two generalizations are totally d-iflferent from the known ones,while they keep the initial form of the classical Pascal's theorem in some degree.Moreover,we provide the algebraic condition for the singularity of a class of spline spaces over a class of more general partitions including Morgan-Scott's type partition.Using the property of hypersurfaces above,the geometric condition is then also derived.