混合有限元法的对偶列式及其应用
Dual Formulations of Mixed Finite Element Methods with Applications
摘要：
混合有限元法使用两个或多个变量求解偏微分方程。离散外微积分理论解释了为什么与不同变量相关的自由度应该存储在原始和对偶域网格上，并使用离散霍奇星在网格之间传递信息。通过分析和算例表明，离散Hodge星的选择对方法的数值稳定性至关重要。此外，我们定义插值函数和离散霍奇星的双重网格，可用于创建以前未考虑的混合方法。从静磁学和达西流的例子进行了详细的研究。
英文摘要：
Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the numerical stability of the method. Additionally, we define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods. Examples from magnetostatics and Darcy flow are examined in detail.
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