An a priori error analysis of adjoint-based super-convergent Galerkin approximations of linear functionals
An a priori error analysis of adjoint-based super-convergent Galerkin approximations of linear functionals
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DOI：
10.1093/imanum/draa102
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发表时间：
2021-01
影响因子：
2.1
通讯作者：
Bernardo Cockburn;S. Xia
中科院分区：
文献类型：
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作者：
Bernardo Cockburn;S. Xia
我们首次对 Cockburn & Wang (2017, Adjoint-based, super-convergent Galerkin approximations of Linear Functions. J. Comput. Sci., 73, 644–666) 中提出的新方法进行先验误差分析，用于计算线性泛函的基于伴随的超收敛 Galerkin 近似。如果 $J(u)$ 是平滑线性函数，其中 $u$ 是稳态扩散问题的解，则标准逼近 $J(u_h)$ 以 $h^{2k+1}$ 阶收敛，其中 $u_h$ 是 $u$ 的可混合不连续伽辽金逼近，且多项式 $k>0$。相比之下，数值实验表明，新方法提供了按 $h^{4k}$ 阶收敛的近似值，并且只需使用计算 $J(u_h)$ 所需计算量的两倍即可完成计算。在这里，我们将这些实验结果置于坚实的数学基础上。我们还展示了旨在探索该方法在理论未涵盖的情况下的收敛特性的数值实验，特别是当解 $u$ 或函数 $J(\cdot )$ 不是很光滑时。最后我们指出如何将这些结果扩展到一般伽辽金方法的情况。
We present the first a priori error analysis of a new method proposed in Cockburn & Wang (2017, Adjoint-based, superconvergent Galerkin approximations of linear functionals. J. Comput. Sci., 73, 644–666), for computing adjoint-based, super-convergent Galerkin approximations of linear functionals. If $J(u)$ is a smooth linear functional, where $u$ is the solution of a steady-state diffusion problem, the standard approximation $J(u_h)$ converges with order $h^{2k+1}$, where $u_h$ is the Hybridizable Discontinuous Galerkin approximation to $u$ with polynomials of degree $k>0$. In contrast, numerical experiments show that the new method provides an approximation that converges with order $h^{4k}$, and can be computed by only using twice the computational effort needed to compute $J(u_h)$. Here, we put these experimental results in firm mathematical ground. We also display numerical experiments devised to explore the convergence properties of the method in cases not covered by the theory, in particular, when the solution $u$ or the functional $J(\cdot )$ are not very smooth. We end by indicating how to extend these results to the case of general Galerkin methods.