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A local projection type stabilization with exponential enrichments applied to one-dimensional advection-diffusion equations

机译:一维对流扩散方程的指数富集局部投影型镇定

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We consider the local projection stabilization (IPS) for solving a singularly perturbed advection-diffusion two-point boundary value problem. In its classical one-level variant, the LPS uses polynomial bubble functions to enrich the standard finite element spaces of continuous, piecewise polynomial functions. As recently shown, the two-level approach can be considered also as a one-level method, however, with piece-wise polynomial enrichments. Here, we study the question under which condition a linearly independent H~1 function can serve as an enrichment for the standard space of continuous, piecewise polynomials of degree r leading to the same type of error estimates for the solution as the original one- and two-level approaches. Moreover, in the constant coefficient case, we derive formulas for the user-chosen stabilization parameter which guarantee that the piecewise linear part of the solution becomes nodal exact. Finally, we choose exponential enrichments based on the asymptotic expansion of the solution and show by numerical tests that compared to the classical one-level variant of the LPS - a considerable improvement of the accuracy of the solution on non-layer adapted meshes can be achieved.
机译:我们考虑使用局部投影稳定化(IPS)来解决奇摄动对流-扩散两点边值问题。在其经典的一级变体中,LPS使用多项式气泡函数来丰富连续的分段多项式函数的标准有限元空间。如最近所示,两级方法也可以被视为单级方法,但是具有分段多项式充实。在这里,我们研究一个问题,在这种情况下,线性独立的H〜1函数可以充实度为r的连续分段多项式的标准空间,从而导致与原始一阶和二阶方程相同的误差估计类型。两级方法。此外,在常数系数的情况下,我们推导了用户选择的稳定参数的公式,这些公式可确保解的分段线性部分变得精确。最后,我们根据解的渐近展开选择指数富集,并通过数值测试表明,与LPS的经典一级变量相比,可以显着提高非分层自适应网格上解的精度。

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