Helmholtz方程有限元方法的精度改进

Helmholtz方程在电磁学、声学等领域的应用都十分广泛,但实际应用中往往不能得出解析解,故现实中常用有限元方法求出高精度的数值解。针对二维Helmholtz方程的性质,分别采用双线性插值和三角插值的方法构造有限元空间的形函数,并推导了刚度矩阵和荷载向量。采用数学软件MATLAB分别做了数值仿真,得出了数值解与解析解之间的误差数据。通过与采用双线性插值构造的有限元空间对比,用数值仿真证明了采用三角插值方法构造有限元空间时,数值解具有更好的精度,且适用于波数较大的情形。

The betterment of precision of finite element method for helmholtz equation

The applications of Helmholtz equation is extensive in electromagnetics, acoustics, and other fields , but in practical ,it is difficult to draw analytical solution, so in reality the commonly used methed is finite element theory to calculate high- precision numerical solution; Based on two-dimensional Helmholtzthe nature of the equations ,we construct a shape function in the finite element space with method of bilinear interpolation and triangular interpolation , and get the stiffness matrix and load vector, meanwhile ,using MATLAB numerical experiment, obtained numerical solutions with the analytical solutionthe error between the data. By comparison of the finite element space constructed with bilinear interpolation, numerical experiments prove that the latter has better accuracy, and applicable in the wave number of the larger situation.