对流-扩散-反应方程界面问题的扩展杂交间断有限元

An extended HDG finite element method for convection-diffusion-reaction equation interface problems

摘要:

本文针对二维和三维对流-扩散-反应方程的界面问题提出了一种基于非贴体网格的扩展杂交间断有限元方法. 该方法在单元的内部分别用分片 k (k≥1) 和 m (m=k,k-1) 次多项式逼近标量函数及其梯度, 在单元边界上用 k 次多项式逼近标量函数的迹, 在界面上则用界面单元内部的 k 次多项式在界面上的限制去逼近标量函数的迹. 对于弱问题, 本文利用 Lax-Milgram 定理证明其解的存在唯一性. 对于离散格式, 本文给出了其解的存在唯一性以及能量范数下的最优误差估计.

Abstract:

This paper proposes an extended hybridizable discontinuous Galerkin (HDG) finite element for the 2D and 3D convection-diffusion-reaction equation interface problems on body-unfitted meshes. This finite element uses piecewise polynomials of degrees k (k≥1) and m (m=k,k-1) to approximate the scalar function and its gradient respectively in the interior of elements, piecewise polynomials of degrees k to approximate the traces of the scalar function on the inter-element boundaries inside the subdomains, and constraints on the interface of piecewise polynomials of degrees k inside interface elements for the traces of the scalar function on the interface. The existence and uniqueness of weak solution for the weak problem and the discrete solution for the discrete scheme are proved respectively. Lax-Milgram theorem is used for the weak problem. The optimal error estimation is derived in the energy norm for the discrete scheme.