Two-grid Methods for Characteristic Finite Volume Element Approximations of Semi-linear Sobolev Equations
Two-grid Methods for Characteristic Finite Volume Element Approximations of Semi-linear Sobolev Equations
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发表时间：
2015
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通讯作者：
Jinliang Yan;Zhiyue Zhang
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作者：
Jinliang Yan;Zhiyue Zhang
本文提出了求解半线性Sobolev方程特征有限体积元的两重网格方法。该方法基于特征线法、两网格法和有限体积单元法。非对称和非线性迭代仅在粗网格(网格大小为H)上执行。而精细网格解(网格大小为h)可以通过一次对称的线性步长得到。结果表明，粗网格可以比细网格粗得多。只要网格尺寸满足h=O(H)，两网格法就能达到渐近最优逼近。因此，求解如此大的半线性索波列夫方程不会比求解一个单一的线性化方程困难得多。
In this paper, two-grid methods for characteristic finite volume element solutions are presented for the semilinear Sobolev equation. The method is based on the methods of characteristics, two-grid method and the finite volume element method. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid size H). And the finegrid solution (with grid size h) can be obtained by a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = O(H). As a result, solving such a large semi-linear Sobolev equations will not be much more difficult than solving one single linearized equation.