A two-grid algorithm for discontinuous Galerkin approximations to nonlinear Sobolev equations is proposed. H¹ norm error estimate of the two-grid method for the nonlinear parabolic problem is derived. The analysis shows that our two-grid discontinuous Galerkin algorithm will achieve asymptotically optimal approximation as long as the mesh sizes satisfy \(h = O(H^{\frac {r+1}{r}})\), where r is the order of the discontinuous finite element space. The numerical experiments are presented to prove the efficiency of our algorithm.

中文翻译：

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非线性Sobolev方程不连续Galerkin逼近的两网格方法

提出了一种用于非线性Sobolev方程的不连续Galerkin逼近的两网格算法。推导了非线性抛物线问题两网格方法的H ¹范数误差估计。分析表明，只要网格尺寸满足\（h = O（H ^ {\ frac {r + 1} {r}}）\），我们的两网格不连续Galerkin算法将实现渐近最优逼近，其中r为不连续有限元空间的顺序。通过数值实验证明了算法的有效性。