A posteriori error estimates for nonlinear problems: finite element discretizations of elliptic equations
A posteriori error estimates for nonlinear problems: finite element discretizations of elliptic equations
复制标题
DOI：
10.2307/2153518
复制
发表时间：
1994-04
影响因子：
2
通讯作者：
R. Verfürth
中科院分区：
文献类型：
--
作者：
R. Verfürth
我们给出了一个一般的框架，推导出后验误差估计的近似解的非线性问题。在第一步中，它证明了近似解的误差可以从上到下由其残差的适当的范数有界。在第二步中，残差的这个范数由残差的合适的有限维近似的类似范数从上和从下界定。这个数量可以很容易地评估，并为许多实际应用尖锐明确的上限和下限很容易获得。然后将一般结果应用于二阶标量拟线性椭圆型偏微分方程的有限元离散、二阶标量线性椭圆型算子的特征值问题以及定常不可压Navier-Stokes方程。他们立即产生的后验误差估计，这可以很容易地计算从给定的数据的问题和计算的数值解，并给出了全球的上限和局部的数值解的误差下界。
We give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems. In a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual. In a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite-dimensional approximation of the residual. This quantity can easily be evaluated, and for many practical applications sharp explicit upper and lower bounds are readily obtained. The general results are then applied to finite element discretizations of scalar quasi-linear elliptic partial differential equations of 2nd order, the eigenvalue problem for scalar linear elliptic operators of 2nd order, and the stationary incompressible Navier-Stokes equations. They immediately yield a posteriori error estimates, which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution.