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-order 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.

提出了带有界面条件强制执行的运动不连续Galerkin有限元方法（LS-MDG-ICE）的最小二乘公式。该方法将MDG-ICE（使用弱公式，分别执行守恒定律和相应的界面条件，并将离散几何作为变量）与Demkowicz和Gopalakrishnan的不连续Petrov-Galerkin（DPG）方法相结合，系统地生成最优方法。离散流场和离散几何的试验空间中的测试函数。对于不粘稠的流动，LS-MDG-ICE可以检测并安装先验未知接口，包括电击。对于以对流为主的扩散，LS-MDG-ICE使用各向异性曲线解析内部层（例如，粘性冲击）和边界层$\mathrm{[R}$-适应性，其中高阶形状表示各向异性地适应以精确解析流场。这样，无论网格分辨率和多项式次数如何，LS-MDG-ICE解决方案都是无振荡的。最后，对于一维线性和非线性问题，LS-MDG-ICE被证明可以实现最优解的收敛。${\mathrm{大号}}^2$ 当离散几何是固定的时，相对于精确解的解误差；当离散几何被视为变量时，是最优收敛。