The Convergence Of Time-Space Continuous Finite Element Method For First Order Hyperbolic Equation

In this paper,four kinds of finite element:biquadratic element;bicubic element and quadratic serendipity element have been used to solve first order hyperbolic equation, moreover,the convergence of quadratic serendipity element has been analyzed theoretically. It shows that the theoretical result is consistent with the numerical result. In addition,the solvability of biquadratic element and quadratic serendipity element has been analyzed in this paper. All the results I got show that continuous finite element method also do well in solving first order hyperbolic equation and first order hyperbolic system. The convergence and superconvergence are very similar with elliptic equation.To linear constant coefficient hyperbolic equations, I use biquadratic element; bicubic element ; quadratic serendipity element to solve the problem respectively. To linear constant (variable) coefficient first order hyperbolic systems I use biquadratic element to solve the problem. All these calculations are in a cell by cell pattern and all results are convergent. However,there are something with bicubic element. I think it need more research. To biquadratic element and quadratic serendipity element there are superconvergence on some points. The theoretical proof of the convergence and superconvergence of quadratic serendipity element was presented in this paper.