扭曲网格上带有非光滑系数扩散方程的加权DG方法  

A weighted discontinuous Galerkin method for diffusion equations with non-smooth coefficients on distorted polygonal meshes

作　　者：张荣培 蔚喜军[2] 李明军[3] Rongpei Zhang;Xijun Yu;Mingjun Li

机构地区：[1]广东工业大学应用数学学院,广州510006 [2]北京应用物理与计算数学研究所计算物理实验室,北京100088 [3]湘潭大学数学与计算科学学院,湘潭411105

出　　处：《中国科学：数学》2022年第7期809-822,共14页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11571002和11971411)资助项目。

摘　　要：具有各向异性和间断扩散系数的椭圆型方程在辐射流体力学和油藏模拟等许多物理应用中发挥着重要作用.辐射扩散问题的计算通常基于流体的网格.在流体计算中,网格会随着流体的流动发生扭曲变形.间断Galerkin(discontinuous Galerkin,DG)方法是计算数学中一类重要方法,适用于间断系数和非规则网格等复杂情形.本文在对称内惩罚方法的基础上发展加权DG方法求解扭曲网格上的椭圆方程.在理论分析中,首先给出DG方法中双线性形式的强制性和连续性的证明,然后基于强制性和连续性给出能量范数的误差分析,最后采用对偶论证技巧给出L^(2)范数下的误差估计.数值实验在随机网格、正弦曲线型网格、Shestakov型网格和Z字型扭曲网格上进行,数值结果验证了加权DG方法对具有间断和各向异性扩散系数的椭圆问题的有效性.The elliptic equations with heterogeneous and anisotropic diffusion coefficients play an important role in many physical applications such as radiation hydrodynamics and reservoir simulations.Usually the distorted meshes are considered since the solution of the diffusion part involves the hydro part.The discontinuous Galerkin(DG)method is an important class of methods in computational mathematics.The features of DG methods can be implemented flexibly on large deformation quadrilateral meshes and discontinuous coefficients.In this study,we present a weighted discontinuous Galerkin(WDG)method for numerically solving the elliptic problem with non-smooth coefficients.The WDG method is based on the symmetric interior penalty technique.We give the proof of the coercivity and continuity of bilinear forms in the WDG scheme.The convergence analysis for the energy norm is presented based on the coercivity and continuity.Next,we prove an error estimate in the L^(2)norm based on the duality argument.Numerical examples demonstrate the validity of the WDG method for elliptic problems with discontinuous and anisotropic diffusion coefficients.Some types of distorted meshes such as random,sinusoidal,Shestakov,and Z meshes are used.

关 键 词：间断GALERKIN方法 扭曲网格 椭圆方程 各向异性 

分 类 号：O241.82[理学—计算数学]