Weighted Error Estimates for Transient Transport Problems Discretized Using Continuous Finite Elements with Interior Penalty Stabilization on the Gradient Jumps
Weighted Error Estimates for Transient Transport Problems Discretized Using Continuous Finite Elements with Interior Penalty Stabilization on the Gradient Jumps
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DOI：
10.1007/s10013-022-00550-x
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发表时间：
2021-04
影响因子：
0.8
通讯作者：
E. Burman
中科院分区：
文献类型：
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作者：
E. Burman
本文考虑一阶标量迁移方程的空间半离散化。对于空间离散化，我们使用标准的连续有限元与稳定组成的罚款跳跃的元素面上的梯度。我们回顾了光滑解和粗糙解的一些全局误差估计，然后证明了瞬态线性迁移方程的一个新的局部误差估计。特别是，我们表明，对于稳定的方法，在解决方案中的非光滑功能的影响呈指数衰减的时空区域的解决方案是粗糙的，使光滑的功能将被运送不受干扰。当精确解是局部光滑的时，局部L2模误差以期望阶收敛。然后，我们用数字说明结果。特别是，我们表现出良好的局部精度在稳定的方法的平滑区域，标准的Galerkin无法近似的解决方案，是顺利的，在最后的时间，如果欠分辨功能已存在于解决方案中的某些时间在进化过程中。
In this paper we consider the semi-discretization in space of a first order scalar transport equation. For the space discretization we use standard continuous finite elements with a stabilization consisting of a penalty on the jump of the gradient over element faces. We recall some global error estimates for smooth and rough solutions and then prove a new local error estimate for the transient linear transport equation. In particular we show that for the stabilized method the effect of non-smooth features in the solution decay exponentially from the space time zone where the solution is rough so that smooth features will be transported unperturbed. Locally theL2-norm error converges with the expected order, if the exact solution is locally smooth. We then illustrate the results numerically. In particular we show the good local accuracy in the smooth zone of the stabilized method and that the standard Galerkin fails to approximate a solution that is smooth at the final time if underresolved features have been present in the solution at some time during the evolution.