sine-Gordon方程的最低阶各向异性混合元高精度分析新途径

在各向异性网格下,针对一类非线性sine-Gordon方程利用最简单的双线性元Q_(11)及Q_(01)×Q_(10)元提出了一个自然满足Brezzi-Babuska条件的最低阶混合元新模式.基于Q_(11)元的积分恒等式结果,建立了插值与Ritz投影之间在H~1模意义下的超收敛估计,再结合关于Q_(01)×Q_(10)元的高精度分析方法和插值后处理技术,对于半离散和全离散格式,均导出了关于原始变量u和流量p=-▽u分别在H~1模和L~2模意义下单独利用插值或Ritz投影所无法得到的超逼近性和超收敛结果.最后,我们对其它一些著名单元也进行了分析,进一步验证了所选单元的合理性和独特优势.

A NEW APPROACH OF THE LOWEST ORDER ANISOTROPIC MIXED FINITE ELEMENT HIGH ACCURACYANALYSIS FOR NONLINEAR SINE-GORDON EQUATIONS

With the help of the simplest bilinear element Q¹¹ and Q⁰¹ × Q¹⁰ element, the lowest order new mixed finite element scheme for nonlinear sine-Gordon equations is proposed, which can satisfy Brezzi-Babu?ka condition automatically on anisotropic meshes. Based on integral indentity result of Q¹¹ element, a superconvergence estimate in H¹-norm is established between the interpolation and Riesz projection, which together with the high accuracy analysis method of Q⁰¹ ×Q¹⁰ element and interpolation post-processing technique can yield the superclose properties and superconvergence results of the original variable u and flux variable p = -?u in H¹-norm and L²-norm for semi-discrete and fully-discrete schemes, which can't be deduced by the interpolation and Riesz projection alone. Finally, we give the analysis of some other famous elements, which shows that the choices of the elements used in the new formulation are reasonable and have distinguish advantages.