A high-order finite element method is proposed to solve the nonlinear convection-diffusion equation on a time-varying domain whose boundary is implicitly driven by the solution of the equation. The method is semi-implicit in the sense that the boundary is traced explicitly with a high-order surface-tracking algorithm, while the convection-diffusion equation is solved implicitly with high-order backward differentiation formulas and fictitious-domain finite element methods. By two numerical experiments for severely deforming domains, we show that optimal convergence orders are obtained in energy norm for third-order and fourth-order methods.
翻译:提议采用高阶有限要素法,在一个时间分配域内解决非线性对流-扩散方程式,其边界由等式的解决方案暗含驱动,该方法半隐含,即边界以高阶地面跟踪算法明确追踪,而对流-扩散方程式则以高阶后向分化公式和虚构-内向有限要素法间接解决。通过两次对严重变形区域进行数字实验,我们显示,第三阶和第四阶方法的能源规范中获得了最佳汇合单。