We use the local orthogonal decomposition technique introduced in Målqvist and Peterseim (Math Comput 83(290):2583–2603, 2014) to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale coefficients. We consider nonsmooth initial data and a backward Euler scheme for the temporal discretization. Optimal order convergence rate, depending only on the contrast, but not on the variations of the coefficients, is proven in the $$L_\infty (L_2)$$L∞(L2)-norm. We present numerical examples, which confirm our theoretical findings.

中文翻译：

---

抛物线方程的多尺度技术

我们使用 Målqvist 和 Peterseim (Math Comput 83(290):2583–2603, 2014) 中引入的局部正交分解技术来推导出具有空间多尺度系数的线性和半线性抛物线方程的广义有限元方法。我们考虑非光滑初始数据和时间离散化的后向欧拉方案。$$L_\infty (L_2)$$L∞(L2)-范数证明了最优阶收敛率，仅取决于对比度，而不取决于系数的变化。我们提供了数值例子，这证实了我们的理论发现。