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.

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\)为\(k\ge 1\)，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\(\varOmega\)是\(\mathbb {R}^2\) 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。