论文标题
带有接口条件执行的不连续的盖尔金有限元法的最小二乘公式
A Least-Squares Formulation of the Moving Discontinuous Galerkin Finite Element Method with Interface Condition Enforcement
论文作者
论文摘要
提出了带有界面条件强制执行（LS-MDG-ICE）的不连续的Galerkin有限元法的最小二乘公式。该方法结合了MDG-ICE，它使用了一种弱公式，该公式分别强制执行保护定律和相应的界面条件，并将离散的几何形状视为变量，与Demkowicz和Gopalakrishnan的不连续的Petrov-Galerkin（DPG）方法论，可以系统地产生DIST型号的最佳测试空间，并在diant flof flof the Trial Flof Spection and Trial Flof tirce from the Trial Flofs and the Trial Flof。对于无粘性流，LS-MDG-ICE检测并适合先验未知的接口，包括冲击。 For convection-dominated diffusion, LS-MDG-ICE resolves internal layers, e.g., viscous shocks, and boundary layers using anisotropic curvilinear $r$-adaptivity in which high-order shape representations are anisotropically adapted to accurately resolve the flow field.因此，无论网格分辨率和多项式程度如何，LS-MDG冰溶液不含振荡。最后，对于一个维度的线性和非线性问题，LS-MDG-ICE被证明可以在固定并将离散几何形状视为可变时，在离散的几何形状固定并且超级最佳几何形状时，相对于精确解决方案获得了$ l^2 $解决方案误差的最佳收敛。
A least-squares formulation of the Moving Discontinuous Galerkin Finite Element Method with Interface Condition Enforcement (LS-MDG-ICE) is presented. This method combines MDG-ICE, which uses a weak formulation that separately enforces a conservation law and the corresponding interface condition and treats the discrete geometry as a variable, with the Discontinuous Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan to systematically generate optimal test functions from the trial spaces of both the discrete flow field and discrete geometry. For inviscid flows, LS-MDG-ICE detects and fits a priori unknown interfaces, including shocks. For convection-dominated diffusion, LS-MDG-ICE resolves internal layers, e.g., viscous shocks, and boundary layers using anisotropic curvilinear $r$-adaptivity in which high-order shape representations are anisotropically adapted to accurately resolve the flow field. As such, LS-MDG-ICE solutions are oscillation-free, regardless of the grid resolution and polynomial degree. Finally, for both linear and nonlinear problems in one dimension, LS-MDG-ICE is shown to achieve optimal convergence of the $L^2$ solution error with respect to the exact solution when the discrete geometry is fixed and super-optimal convergence when the discrete geometry is treated as a variable.
