A Collocation Finite Element Method For Elliptic Problems In Irregular Domain

The collocation method is a munerical method which searches for the approximation solution of the operator function by satisfying pure interpolation condition for about thirty years,and it is widely used for solving both engineering and computing mathematics due to its ease of implementation,high-order accuracy and no integrals need be evaluated.Collocation method essentially involves determining an approximate solution by a piecewise polynomial by requiring it to satisfy the differential equation and boundary conditions exactly at certain points.Original spline collocation methods collocate at the nodes by cubic spline functions,but the precision isn't good.For improving the convergence rate,collocation points usually use the nodes of Gauss quadrature formula,and choose piecewise Hermite bicubic polynomials as the approximative space,convergence rate can reach h~4,spline collocation at Gauss points is named orthogonal spline collocation method(OSC).Orthogonal collocation method was first introduced by C.deBoor and Swartz[2] for m-th order ODE.In one space variable,Douglas and Dupont[3]give C~1 finite element method(R≥3).In two space variables,Prenter and Rusell[6]gives OSC for elliptic equation.Bialecki and Cai[11]consider two kinds of interpolation for the boundary conditions of elliptic equation,i.e.Hermite intetpolation and Gauss interpolation, optimal estimate can be get.Percell and Wheeler[5]consider the elliptic problems(R≥3).Bialeki[12]extends and generalizes the theoretical result for elliptic problems,and in[13]gets superconvergence result.Orthogonal collocation method is higher convergence rate than finite element ,since OSC needn't compute integrals,numerical integrals increase the workload and effect the precision of coefficient matrix.Then collocation methods are used widely for mathematical physics snd engineering problems.Emil O.Frind and George F.Pinder(?)1 gave a collocation finite element method for Laplace equation in irregular domains.The main work of this thesis is to extend this method to elliptic problems.In this thesis,a potentially powerful numerical method for solving elliptic boundary value problems in irregular domains is proposed.The method combines the simplicity of orthogonal collocation with the versatility of deformable finite elements.Bicubic Hermite elements with four degreesof-freedom per node are used.A subparametric transformation permits the precise positioning of the collocation points for maximum accuracy as well as a unique representatin of irregular boundaries.This thesis is divided into three chapters.In chapter one,we give base theory of collocation element,method for Laplace equation in irregular domains and give three lemmas.In chapter two,we give collocation finite element method for elliptic equation in irregular domains:In chapter three,we give two numerical examples,and give a comparison between orthogonal collocation and Galerkin finite elements in computational efficiency.In the end,we give the conclusion and the shortage of this thesis.