Bivariate Splines Finite Element Solver For Linear Hyperbolic Equations In Two Space Dimension

The hyperbolic equation reflects the wave phenomenon in nature.The study of such problems is related to many practical problems,such as string vibration,elastic film or three-dimensional elastic vibration,in describing the relationship between electric field,magnetic field and charge density,current density,have important meaning.Spline is a special function defined by polynomial segmentation.In computer science cad and computer graphics,splines usually refer to piecewise defined polynomial parametric curves.Because the spline is simple in construction,easy to use,accurate in fitting,and able to approximate complex shapes in curve fitting and interactive curve design,splines are a common representation of curves in these fields.In this paper,the binary spline theory is combined with the classical finite element method,and a numerical method for solving linear hyperbolic equations in two-dimensional space is proposed.The method is based on 2-type triangulation and satisfies homogeneous boundary condition by constructing b-spline interpolation boundary function.At the same time,the spline method uses the binary spline basis function to construct the understanding space and the test space,thus effectively solving the two-dimensional linear hyperbolic equation.Compared with the traditional finite element method,the method has higher precision and more importantly,the method does not need to divide the spatial variables.Then two examples are used to confirm the correctness of the theoretical conclusion,and it is found that the approximate solution is very consistent with the known exact solution.This means that the spline method is effective and feasible for solving two-dimensional linear hyperbolic equations.The structure of this article is as follows:The first chapter is the introduction,which briefly introduces the background significance and research status of hyperbolic equation.In the second chapter,we discuss the binary spline space,and mainly introduce the spline function space on 2-type triangulation,because of its simple structure and good symmetry,it is widely used in practice.The third chapter discusses the numerical solution of the two-dimensional linear hyperbolic equation.Firstly,the classical finite element method is introduced.The method uses the Galerkin method to discretize in the time direction and constructs the finite element in space.In chapter four,two numerical examples are given to verify the feasibility of this method.