Gauss–Newton Multilevel Methods for Least-Squares Finite Element Computations of Variably Saturated Subsurface Flow
Gauss–Newton Multilevel Methods for Least-Squares Finite Element Computations of Variably Saturated Subsurface Flow
复制标题
变饱和地下流最小二乘有限元计算的高斯-牛顿多级方法
作者：
G. Starke
摘要 我们将最小二乘混合有限元框架应用于变饱和流隐式欧拉离散化的每个时间步中出现的非线性椭圆问题。这种方法允许将符合标准分段线性 H1 的水力势有限元与符合 H(div) 的通量 Raviart-Thomas 空间相结合。它还提供了可用于自适应网格细化策略的后验误差估计器。由此产生的非线性代数最小二乘问题通过不精确的高斯-牛顿方法求解，该方法使用内部迭代的停止准则，该准则基于线性化最小二乘函数相对于非线性最小二乘函数的变化。内部迭代是使用自适应多级方法和块高斯-赛德尔平滑迭代来执行的。对于实际的地下水位补给问题，给出了计算实验的结果。
Abstract We apply the least-squares mixed finite element framework to the nonlinear elliptic problems arising in each time-step of an implicit Euler discretization for variably saturated flow. This approach allows the combination of standard piecewise linear H1-conforming finite elements for the hydraulic potential with the H(div)-conforming Raviart–Thomas spaces for the flux. It also provides an a posteriori error estimator which may be used in an adaptive mesh refinement strategy. The resulting nonlinear algebraic least-squares problems are solved by an inexact Gauss–Newton method using a stopping criterion for the inner iteration which is based on the change of the linearized least-squares functional relative to the nonlinear least-squares functional. The inner iteration is carried out using an adaptive multilevel method with a block Gauss–Seidel smoothing iteration. For a realistic water table recharge problem, the results of computational experiments are presented.