Abstract

We deal with the numerical solution of linear partial differential equations (PDEs) with focus on the goal-oriented error estimates including algebraic errors arising by an inaccurate solution of the corresponding algebraic systems. The goal-oriented error estimates require the solution of the primal as well as dual algebraic systems. We solve both systems simultaneously using the bi-conjugate gradient method which allows to control the algebraic errors of both systems. We develop a stopping criterion which is cheap to evaluate and guarantees that the estimation of the algebraic error is smaller than the estimation of the discretization error. Using this criterion and an adaptive mesh refinement technique, we obtain an efficient and robust method for the numerical solution of PDEs, which is demonstrated by several numerical experiments.

中文翻译：

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面向目标的误差估计中线性代数系统的有效数值解

摘要

我们处理线性偏微分方程（PDE）的数值解，重点是面向目标的误差估计，包括由相应代数系统的不精确解引起的代数误差。面向目标的误差估计需要求解原始代数系统和对偶代数系统。我们使用双共轭梯度法同时求解两个系统，该方法可控制两个系统的代数误差。我们开发了一种停止准则，该准则的评估成本较低，并且可以保证代数误差的估计小于离散化误差的估计。使用该准则和自适应网格细化技术，我们获得了一种有效且鲁棒的PDE数值解方法，该方法已通过若干数值实验得到了证明。