Newton-Krylov-Schwarz Methods

Newton-Krylov-Schwarz (NKS) methods use a Krylov subspace methods to solve for the Newton step. Schwarz preconditioners are used to accelerate the convergence of the Krylov method. We are interested in developing numerical solutions for nonlinear, temporally dependent partial differential equations. We advance the systems in time using an NKS method.

My particular research focus on this project has been the development of preconditioners for our problems. We use domain decomposition preconditioners because we want to run simulations on massively parallel machines. We have found that two-level methods significantly reduce iteration counts for the Krylov method, making the development of a coarse mesh problem is crucial.

The Adaptive Hydrology (ADH) Model is a finite-element based code developed at the Engineer Research and Development Center (ERDC). This code solves problems in groundwater, surface water, contaminant transport and rainfall runoff, and also integrates models to solve coupled systems. The solver technology thus needs to be extremely robust. The meshes are three-dimensional and unstructured, so that development of an independent coarse mesh problem is extremely difficult. An independent problem would require generation of a coarse mesh, discretization, parallelization, solution on the mesh, and interpolation between the two meshes.

Alternatively, one can use ideas from algebraic multigrid to form the coarse mesh problem in terms of the existing fine mesh basis functions (and hence the fine mesh matrix). We chose to use aggregation to form the problem and found this to be easy and extremely effective. We were able to reduce the iteration counts in our test problems by a factor of 10, and by lagging the coarse mesh problem over several Newton step were able to significantly reduce computational times.

This animation shows changing values in saturation as an initially saturated soil column is allowed to drain. The soil column is made of sand, silt, and clay layers, which have varying conductivity properties. The data for the animations were generated using the ADH Model, and the graphics were developed using the Groundwater Modeling System, or GMS.

The initial value of saturation in the column is 1.0, meaning the the pore spaces in the material in the column are completely filled with water. You can see the values of saturation changing as the column is drained (where red is 0.0 saturation). Notice how the material properties affect the drainage.

A second animation shows the corresponding values of total head for the same column experiment. This second animation shows the values of the total (or hydraulic) head calculation given by ADH. Initially, the entire column has a value of 0.0 (blue) for the total head. The total head may be thought of as a normalized energy. Once the support of the water in the column is removed, the value of the total head at the bottom is -10 (red). You can see that it takes longer for the total head to propagate through the clay lenses due to the lower conductivity values of this material. At the end, the top of the column has a total head value close to $-4$, indicating that there is still a head gradient, which will continue to induce flow.