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A weighted and balanced FEM for singularly perturbed reaction-diffusion problems
Calcolo ( IF 1.4 ) Pub Date : 2021-06-06 , DOI: 10.1007/s10092-021-00421-w
Niall Madden , Martin Stynes

A new finite element method is presented for a general class of singularly perturbed reaction-diffusion problems \(-\varepsilon ^2\varDelta u +bu=f\) posed on bounded domains \(\varOmega \subset \mathbb {R}^k\) for \(k\ge 1\), with the Dirichlet boundary condition \(u=0\) on \(\partial \varOmega\), where \(0 <\varepsilon \ll 1\). The method is shown to be quasioptimal (on arbitrary meshes and for arbitrary conforming finite element spaces) with respect to a weighted norm that is known to be balanced when one has a typical decomposition of the unknown solution into smooth and layer components. A robust (i.e., independent of \(\varepsilon\)) almost first-order error bound for a particular FEM comprising piecewise bilinears on a Shishkin mesh is proved in detail for the case where \(\varOmega\) is the unit square in \(\mathbb {R}^2\). Numerical results illustrate the performance of the method.



中文翻译:

奇异扰动反应扩散问题的加权平衡有限元法

提出了一种新的有限元方法,用于在有界域\(\varOmega \subset \mathbb {R}^ )上提出的一类奇异摄动反应扩散问题\(-\varepsilon ^2\varDelta u +bu=f\) k\)\(k\ge 1\),狄利克雷边界条件\(u=0\)在 \(\partial \varOmega\) 上,其中\(0 <\varepsilon \ll 1\)。该方法被证明是准最优的(在任意网格和任意一致的有限元空间上)关于加权范数,当人们将未知解典型地分解为平滑和层分量时,该范数已知是平衡的。一个健壮的(即,独立于\(\varepsilon\)) 包含 Shishkin 网格上的分段双线性的特定 FEM 的几乎一阶误差界限在\(\varOmega\)\(\mathbb {R}^2\) 中的单位平方的情况下得到了详细证明。数值结果说明了该方法的性能。

更新日期:2021-06-07
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