标题： 一种用于对流扩散问题不连续伽辽金离散化的后验误差估计器及其在地球地幔对流模拟中的应用 显示英文标题

标题： An posteriori error estimator for discontinuous Galerkin discretisations of convection-diffusion problems with application to Earth's mantle convection simulations

摘要： 我们为非定常对流-扩散方程应用的内部惩罚不连续伽辽金方法提出了新的后验误差估计。 重点是强对流占主导的问题，没有零阶反应项，这导致了正的L^2类似分量的缺失。 一个重要的具体例子是来自地球地幔对流建模的Boussinesq系统的能量/温度方程。 通过一种指数拟合技术来解决由于使用Gronwall类型论证而产生的关于最终时间的指数因子的影响这一关键数学挑战。 后者得到一类新的针对稳态问题的后验误差估计，在现有文献未涵盖的对流和反应系数组合情况下有效。 这种新的估计器与椭圆重构技术相结合，推导出针对非稳态问题的新相应估计，表现出对Gronwall类型指数的依赖性降低，从而在更长的时间区间内提供更精确的估计。 我们在地球地幔对流模拟中展示了新一类后验误差估计器在驱动网格自适应方面的优越性能，在该设置中，能量/温度方程由不连续伽辽金方法离散化，并与Taylor-Hood有限元方法耦合用于动量和质量守恒方程。 我们利用社区代码ASPECT，展示所提出方法的效果性数值例子。 显示英文摘要

摘要： We present new aposteriori error estimates for the interior penalty discontinuous Galerkin method applied to non-stationary convection-diffusion equations. The focus is on strongly convection-dominated problems without zeroth-order reaction terms, which leads to the absence of positive L^2-like components. An important specific example is the energy/temperature equation of the Boussinesq system arising from the modelling of mantle convection of the Earth. The key mathematical challenge of mitigating the effects of exponential factors with respect to the final time, arising from the use of Gronwall-type arguments, is addressed by an exponential fitting technique. The latter results to a new class of aposteriori error estimates for the stationary problem, which are valid in cases of convection and reaction coefficient combinations not covered by the existing literature. This new class of estimators is combined with an elliptic reconstruction technique to derive new respective estimates for the non-stationary problem, exhibiting reduced dependence on Gronwall-type exponents and, thus, offer more accurate estimation for longer time intervals. We showcase the superior performance of the new class of aposteriori error estimators in driving mesh adaptivity in Earth's mantle convection simulations, in a setting where the energy/temperature equation is discretised by the discontinuous Galerkin method, coupled with the Taylor-Hood finite element for the momentum and mass conservation equations. We exploit the community code ASPECT, to present numerical examples showing the effectivity of the proposed approach.