论文标题
混合$（\ MATHCAL {l} \ MATHCAL {l}^*）^{ - 1} $和$ \ MATHCAL {l} \ MATHCAL {L}^*$最少squares有限元元素方法与application applistion to Linareareareareareareareartarcolic问题
Mixed $(\mathcal{L}\mathcal{L}^*)^{-1}$ and $\mathcal{L}\mathcal{L}^*$ least-squares finite element methods with application to linear hyperbolic problems
论文作者
论文摘要
在本文中，研究了一些双重最小二乘有限元方法及其在标量线性双曲线问题上的应用。目的是在精确解决方案的有限元元素上获得$ l^2 $ norm近似值，以获得双曲线偏微分方程的关注。通过使用原始的$ \ Mathcal {l} \ Mathcal {l} \ Mathcal {L} \ Mathcal {l}^*$方法提出，在Elliptic eartiptic eartiptic eartiptic eartiptication的背景下，使用原始的最小二乘原理来定义了精确解决方案的$ l^2 $ - 正交投影，从而定义了精确解决方案的$ l^2 $ - 正交投影，从而定义了这一点。本文中的所有方法均建立并扩展$ \ MATHCAL {L} \ MATHCAL {l}^*$方法，这在椭圆形问题的设置之外相当一般且适用。误差界表明，指向影响收敛的因素并提供保证最佳速率的条件。此外，讨论了所得线性系统的预处理。提供数值结果以说明在共同有限元空间上的方法的行为。
In this paper, a few dual least-squares finite element methods and their application to scalar linear hyperbolic problems are studied. The purpose is to obtain $L^2$-norm approximations on finite element spaces of the exact solutions to hyperbolic partial differential equations of interest. This is approached by approximating the generally infeasible quadratic minimization, that defines the $L^2$-orthogonal projection of the exact solution, by feasible least-squares principles using the ideas of the original $\mathcal{L}\mathcal{L}^*$ method proposed in the context of elliptic equations. All methods in this paper are founded upon and extend the $\mathcal{L}\mathcal{L}^*$ approach which is rather general and applicable beyond the setting of elliptic problems. Error bounds are shown that point to the factors affecting the convergence and provide conditions that guarantee optimal rates. Furthermore, the preconditioning of the resulting linear systems is discussed. Numerical results are provided to illustrate the behavior of the methods on common finite element spaces.
