In this paper, a weak Galerkin finite element method is proposed and analyzed for one-dimensional singularly perturbed convection–diffusion problems. This finite element scheme features piecewise polynomials of degree \(k\ge 1\) on interior of each element plus piecewise constant on the node of each element. Our WG scheme is parameter-free and has competitive number of unknowns since the interior unknowns can be eliminated efficiently from the discrete linear system. An \(\varepsilon \)-uniform error bound of \(\mathcal {O}((N^{-1}\ln N)^k)\) in the energy-like norm is established on Shishkin mesh, where N is the number of elements. Finally, the numerical experiments are carried out to confirm the theoretical results. Moreover, the numerical results show that the present method has the optimal convergence rate of \(\mathcal {O}(N^{-(k+1)})\) in the \(L^2\)-norm and the superconvergence rates of \(\mathcal {O}((N^{-1}\ln N)^{2k})\) in the discrete \(L^{\infty }\)-norm.

本文针对一维奇异摄动对流扩散问题，提出了一种弱Galerkin有限元方法并对其进行了分析。此有限元方案的特征是每个元素内部的度为\（k \ ge 1 \）的分段多项式加上每个元素节点上的分段常数。由于可以从离散线性系统中有效地消除内部未知数，因此我们的WG方案没有参数，并且具有未知数量的竞争性。在Shishkin网格上建立了类能量范数中\（\ mathcal {O}（（N ^ {-1} \ ln N）^ k）\）的\（\ varepsilon \）均匀误差界线，其中N是元素的数量。最后，进行了数值实验，以验证理论结果。此外，数值结果表明，该方法在\（L ^ 2 \）-范数下具有最优的收敛速度\（\ mathcal {O}（N ^ {-（k + 1）}）\）。离散\（L ^ {\ infty} \）-范数中\（\ mathcal {O}（（N ^ {-1} \ ln N）^ {2k}）\）的超收敛速率。