应力张量强对称Stokes流的超收敛HDG方法
A superconvergent HDG method for Stokes flow with strongly enforced symmetry of the stress tensor
摘要：
本文提出了一种超收敛的杂交间断Galerkin（HDG）方法来逼近斯托克斯方程的Cauchy形式，对原始变量和混合变量使用相同的多项式的次数.新的配方依赖于著名的福格特符号，以强烈执行应力张量的对称性。所提出的策略相对于现有的HDG制剂引入了几个优点。首先，它补救了经典HDG方法所经历的次优行为，涉及原始变量的梯度的对称部分的配方。混合变量的最佳收敛检索和元素的元素后处理程序导致超收敛速度场，即使是低阶近似。第二，不需要额外的离散空间的富集，并且计算效率的增益来自于减少存储的信息的量和局部问题的大小。最终，新的公式自然地对诺依曼边界施加物理牵引力。数值验证的方法的最优性和超收敛性能进行在2D和3D使用不同的元素类型的网格。
英文摘要：
This work proposes a superconvergent hybridizable discontinuous Galerkin (HDG) method for the approximation of the Cauchy formulation of the Stokes equation using same degree of polynomials for the primal and mixed variables. The novel formulation relies on the well-known Voigt notation to strongly enforce the symmetry of the stress tensor. The proposed strategy introduces several advantages with respect to the existing HDG formulations. First, it remedies the suboptimal behavior experienced by the classical HDG method for formulations involving the symmetric part of the gradient of the primal variable. The optimal convergence of the mixed variable is retrieved and an element-by-element post-process procedure leads to a superconvergent velocity field, even for low-order approximations. Second, no additional enrichment of the discrete spaces is required and a gain in computational efficiency follows from reducing the quantity of stored information and the size of the local problems. Eventually, the novel formulation naturally imposes physical tractions on the Neumann boundary. Numerical validation of the optimality of the method and its superconvergent properties is performed in 2D and 3D using meshes of different element types.
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