The Q₁-Mortar Finite Element Method For Poisson Equation

In this thesis, we considered a discretization of linear elliptic boundary value prob-lem in 2-D by the new version of the Mortar finite element method which uses locallynonconforming Q1-Mortar finite elements. Mortar finite element appeared first in 1994.After more than ten years of development, Mortar finite element has been used in manyareas, such as machine building industry, biomolecule etc. The characteristic of theMortar finite element method is coordinate the mesh over two separate componentssuch that they are conforming at interfaces. So it has usually been used to solve PDEon a complex area. And with the diversification of algorithm, there have been a seriesof solving PDE with Mortar finite element method.Usual Mortar finite element method uses the locally conforming elements on thesubdomains, but we considered a Mortar element with locally nonconforming Q1 el-ements, also called the rotation Q1 elements. On the subdomains, we used di?erentsizes of rectangles of subdivision for solving the Poisson equation, only one Mortarcondition is required in the case, i.e. the traces of solutions of the two neighboringsubdomains have the same L2 projections on the Mortar space at the interfaces. Butfor other di?erent finite elements, the Mortar condition needed is also di?erent. Forexample, there are two Mortar conditions for the Morley finite element. In generally,the Mortar condition needed is to keep the global convergence.After detailed analysis, we proved that our error estimate is as the same optimalorder as in the standard linear nonconforming finite element method. To verify thetheoretical result, we also gave the corresponding numerical examples.