An hp-Adaptive Scheme of Discontinuous Galerkin Time-Domain Method With Fast Error Estimation
Yan Shi, Peng Wang, Shi Chen Zhu, Shuai Peng Li, Zhen Guo Ban
Abstract
In this article, a highly efficient <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hp</i> -adaptive scheme has been developed for the discontinuous Galerkin time-domain (DGTD) method. In order to achieve the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${h}$ </tex-math></inline-formula> - and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${p}$ </tex-math></inline-formula> -adaptive behaviors, a hierarchical octree-based recursive meshing procedure has been introduced. In the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hp</i> -adaptive scheme, the transformation method of the basis function is developed to convert the fields between the parent element and the child elements and between the higher- and the lower-order bases. Instead of the time-consuming field update process according to the governing equations, the error based on modified reference solutions can be rapidly and accurately calculated in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${h}$ </tex-math></inline-formula> - and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${p}$ </tex-math></inline-formula> -adaptions by using the proposed transformation, thus flexibly manipulating the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hp</i> -adaptive procedure. Some 3-D radiation and scattering examples are given to demonstrate the good accuracy and high efficiency of the proposed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hp</i> -adaptive DGTD method.