论文标题
自动变异稳定的Fe方法的目标误差估计，以对流为主导的扩散问题
Goal-Oriented Error Estimation for the Automatic Variationally Stable FE Method for Convection-Dominated Diffusion Problems
论文作者
论文摘要
我们提出了针对目标变异稳定有限元（AVS FE）方法的后验误差估计值，以定位标量值对流 - 对流 - 扩散问题。 AVS-FE方法是一种破坏测试空间的Petrov-Galerkin方法，而试验空间由经典的Fe基础函数组成，例如C0或Raviart Thomas函数。我们采用了Demkowicz和Gopalakrishnan的不连续Petrov Galerkin（DPG）方法的最佳测试功能的概念，从而实现了无条件稳定的Fe近似值。值得注意的是，通过使用C0或Raviart Thomas试验空间，可以通过元素方式以完全脱钩的元素来计算最佳的不连续测试功能。 为了建立误差估计器，我们提出了两种方法：i）遵循贝克尔和兰纳切尔的经典方法，即，在（破碎的）测试空间中寻求双重解决方案，以及ii），引入了我们寻求C0或Raviart Thomas，AVS Fe fe解决方案的替代方法，AVS Fe解决方案通过使用基础强的强度形式的强度界数（Bvp）（bvp）（Bvp）（Bvp）。 2D对流为主导的扩散BVP的各种数值验证表明，通过新的替代方法对近似误差的估计值高度准确，而经典方法导致质量差的误差估计值。最后，我们通过新的替代方法基于对数值近似误差的控制，为H自适应过程提供了一种算法。数值验证表明，随着误差收敛到零，估计器保持高精度。
We present goal-oriented a posteriori error estimates for the automatic variationally stable finite element (AVS FE) method for scalar-valued convection-diffusion problems. The AVS-FE method is a Petrov-Galerkin method in which the test space is broken, whereas the trial space consists of classical FE basis functions, e.g., C0 or Raviart Thomas functions. We employ the concept of optimal test functions of the discontinuous Petrov Galerkin (DPG) method by Demkowicz and Gopalakrishnan, leading to unconditionally stable FE approximations. Remarkably, by using C0 or Raviart Thomas trial spaces, the optimal discontinuous test functions can be computed in a completely decoupled element by element fashion. To establish the error estimators we present two approaches: i) following the classical approach of Becker and Rannacher, i.e., the dual solution is sought in the (broken) test space, and ii) introducing an alternative approach in which we seek C0, or Raviart Thomas, AVS FE approximations of the dual solution by using the underlying strong form of the dual boundary value problem (BVP). Various numerical verifications for 2D convection-dominated diffusion BVPs show that the estimates of the approximation error by the new alternative method are highly accurate, while the classical approach leads to error estimates of poor quality. Lastly, we present an algorithm for h adaptive processes based on control of the numerical approximation error via the new alternative approach. Numerical verifications show that the estimator maintains high accuracy as the error converges to zero.
