论文标题
沿着沿着沿着随机二级椭圆方程的最佳有限元素
Optimal finite elements for ergodic stochastic two-scale elliptic equations
论文作者
论文摘要
我们开发了一种基本最佳的有限元方法，用于求解千古的随机二级椭圆方程，其两尺度系数也可能取决于缓慢的变量。我们解决了从均值中获得的有限的随机二级均等方程式（A. Bourgeat，A。Mikelic和S. Wright，J。ReineAngew。Math，第456页，1994年），其解决方案由跨度的跨度和正确的构造构成的跨度变化构成，并构成了快速构成的态度，该解决方案构成了快速构成的构成，并构成了快速构成的构成，并构成了快速构成的构成，该解决方案构成了快速构成的态度。元素。我们表明，截断水平的收敛速率等于解决同一截断域中的细胞问题的收敛速率。解决该方程式，我们同时获得了对均质方程和校正器的解决方案，仅使用多个自由度，这些自由度基本上等同于解决一个细胞问题所需的自由度。当校正器相对于快速变量和慢速变量都具有足够的规律性时，将获得最佳的复杂性。尽管校正器的规范性规范取决于截短域的大小，但我们表明，溶液对均质方程的近似的收敛速率与截面域的大小无关。借助分析校正器的可用性，我们为原始随机二级方程的解构建了一个数值校正器，从有限元溶液到截断的随机二尺度均质均质方程。准周期两级方程的数值示例，以及Checker板类型的随机两尺度方程（其系数是不连续的），确认了理论结果。
We develop an essentially optimal finite element approach for solving ergodic stochastic two-scale elliptic equations whose two-scale coefficient may depend also on the slow variable. We solve the limiting stochastic two-scale homogenized equation obtained from the stochastic two-scale convergence in the mean (A. Bourgeat, A. Mikelic and S. Wright, J. reine angew. Math, Vol. 456, 1994), whose solution comprises of the solution to the homogenized equation and the corrector, by truncating the infinite domain of the fast variable and using the sparse tensor product finite elements. We show that the convergence rate in terms of the truncation level is equivalent to that for solving the cell problems in the same truncated domain. Solving this equation, we obtain the solution to the homogenized equation and the corrector at the same time, using only a number of degrees of freedom that is essentially equivalent to that required for solving one cell problem. Optimal complexity is obtained when the corrector possesses sufficient regularity with respect to both the fast and the slow variables. Although the regularity norm of the corrector depends on the size of the truncated domain, we show that the convergence rate of the approximation for the solution to the homogenized equation is independent of the size of the truncated domain. With the availability of an analytic corrector, we construct a numerical corrector for the solution of the original stochastic two-scale equation from the finite element solution to the truncated stochastic two-scale homogenized equation. Numerical examples of quasi-periodic two-scale equations, and a stochastic two-scale equation of the checker board type, whose coefficient is discontinuous, confirm the theoretical results.
