The Lumped Mass Finite Element Method For Second Order Hyperbolic Equtions

Most of hyperbolic equations represent wave equations, they are used to describedi?erent waves in physics. Therefore,it is important to study the numerical computationmethods of hyperbolic equations . So far, there are many numerical methods for the relatedtopics, such as ADI (Alternating Direction Implicit) method in di?erence method,GalerkinFEM (Finite Element Method) and LMFEM (Lumped Mass Finite Element Method)infinite element method.But, when the LMFEM is used to solve hyperbolic equation, forsolving the second derivate of time, the first derivate of time is usually introduced as anintermediate variable for constructing a double fully discrete scheme. In this paper,a threelayer fully discrete lumped mass finite element method is developed to solve two kinds ofsecond-order hyperbolic equations,specially,with the constant cod?cient hyperbolic equa-tion and the variable coe?cient hyperbolic equation.The main work of this paper includes the following sections:(1) Summarize the origin, development and applied areas of FEM as well as its problemsolving progress; introduce the essence of LMFEM and its advantages comparing withcommon finite element method.(2) Discuss the LMFEM of second-order hyperbolic equation with constant coe?cientsbase on reference [5-6]. Firstly, give the semi-discrete lumped mass coordination finiteelement method of hyperbolic equation with constant coe?cients. Secondly, establish athree layer fully discrete lumped mass coordination finite element method and prove its reliability by giving a numerical experiment on the basis of semi-discretization.(3) Establish a three layer fully discrete lumped mass non-conforming finite elementmethod for the second-order hyperbolic equation with variable coe?cient and analyse itserror, by using the coordination finite element space constructed in reference [7].