Crosswind Smear and Pointwise Errors in Streamline Diffusion Finite Element Methods
Crosswind Smear and Pointwise Errors in Streamline Diffusion Finite Element Methods
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DOI：
10.1090/s0025-5718-1987-0890252-8
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发表时间：
1987-09
影响因子：
2
通讯作者：
Claes Johnson;A. Schatz;L. Wahlbin
中科院分区：
文献类型：
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作者：
Claes Johnson;A. Schatz;L. Wahlbin
对于一个对流占优的奇异摄动对流扩散问题，通过引入合理的人工对流扩散，使数值流线扩散有限元法中的对流扩散拖尾最小化.随后的方法与分段线性元素收敛的逐点精度几乎h 5/4下的局部光滑性假设。1.导论.流线扩散法是求解对流占优的对流扩散问题的一种有限元方法，它具有形式上的高精度和良好的稳定性。休斯和布鲁克斯（7）在稳态问题中介绍了该方法，参见。Raithby和Torrance（14）和Wahlbin（15）在这个方向上的早期思想。该方法的数学分析始于约翰逊和Navert（8），并继续扩展到，例如，Navert（12）、约翰逊、Navert和Pitkaranta（9）以及约翰逊和Saranen（10）中的时间相关问题。在这些文件中的局部误差估计在L2的顺序O（hk 1/2），在光滑的区域，分段多项式有限元的次数k，推导出，连同估计指出，作为一个典型的例子，在零扩散限制的尖锐不连续性的精确解跨流线将被捕获在一个数值的边界层的宽度为0（h1/2），基本上。本文的目的是首先将上述数值风迹的结果改进为0（h ~ 3/4）。通过在该方法中加入少量的0（h ~ 3/2）人工散射，获得了从0（h ~（1/2））到0（h ~（3/4））的改进。在分段线性的情况下（k = 1），这不会破坏已知的O（h3/2）的精度在L2中的光滑区域。使用我们的第一个结果，然后我们得到我们的第二个主要结果，局部逐点误差估计的顺序0（h5/4）的光滑区域。(The在分段线性情况下，先前已知的最佳逐点误差估计是0（h1/2）。另一个结果是一个全球L，估计的顺序0（h '/2）在典型的迎风和顺风奇点的存在。我们将考虑求u = u（x，y）的模型问题，使得
For a model convection-dominated singularly perturbed convection-diffusion prob- lem, it is shown that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion. The ensuing method with piecewise linear elements converges with a pointwise accuracy of almost h 5/4 under local smoothness assumptions. 1. Introduction. The streamline diffusion method is a finite element method for convection-dominated convection-diffusion problems which combines formal high accuracy with decent stability properties. The method was introduced in the case of stationary problems by Hughes and Brooks (7), cf. Raithby and Torrance (14) and Wahlbin (15) for earlier thoughts in this direction. The mathematical analysis of the method was started in Johnson and Navert (8) and continued with extensions to, e.g., time-dependent problems in Navert (12), Johnson, Navert and Pitkaranta (9) and Johnson and Saranen (10). In these papers local error estimates in L2 of order O(h k 1/2), in regions of smoothness, with piecewise polynomial finite elements of degree k, were derived, together with estimates stating, as a typical example, that in the zero diffusion limit a sharp discontinuity in the exact solution across a streamline will be captured in a numerical crosswind layer of width 0(h1/2), essentially. The purpose of the present paper is first to improve the result just mentioned on numerical crosswind smear to 0(h3/4). The improvement from 0(h1/2) to 0(h3/4) is obtained by adding a small amount, 0(h3/2), of artificial crosswind diffusion to the method. In the piecewise linear case (k = 1) this does not destroy the known O( h3/2) accuracy in L2 in smooth regions. Using our first result, we then obtain our second main result, localized pointwise error estimates of order 0(h5/4) in regions of smoothness. (The previously known best pointwise error estimate in the piecewise linear situation is 0(h1/2).) Another consequence is a global L,-estimate of order 0(h'/2) in the presence of typical crosswind and downwind singularities. We shall consider the model problem of finding u = u(x, y) such that