Abstract Local discontinuous Galerkin method is considered for time-dependent singularly perturbed semilinear problems with boundary layer. The method is equipped with a general numerical flux including two kinds of independent parameters. By virtue of the weighted estimates and suitably designed global projections, we establish optimal ( k + 1 ) {(k+1)} -th error estimate in a local region at a distance of 𝒪 ⁢ ( h ⁢ log ⁡ ( 1 h ) ) {\mathcal{O}(h\log(\frac{1}{h}))} from domain boundary. Here k is the degree of piecewise polynomials in the discontinuous finite element space and h is the maximum mesh size. Both semi-discrete LDG method and fully discrete LDG method with a third-order explicit Runge–Kutta time-marching are considered. Numerical experiments support our theoretical results.

中文翻译：

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瞬态奇异摄动半线性反应扩散问题的局部不连续伽辽金方法

摘要 考虑局部不连续伽辽金法求解具有边界层的瞬态奇异摄动半线性问题。该方法配备了包括两种独立参数的通用数值通量。凭借加权估计和适当设计的全局投影，我们在距离 𝒪 ⁢ ( h ⁢ log ⁡ ( 1 h ) 的局部区域中建立了最优 ( k + 1 ) {(k+1)} -th 误差估计) {\mathcal{O}(h\log(\frac{1}{h}))} 来自域边界。这里 k 是不连续有限元空间中分段多项式的次数，h 是最大网格尺寸。考虑了半离散 LDG 方法和具有三阶显式 Runge-Kutta 时间推进的完全离散 LDG 方法。数值实验支持我们的理论结果。