In this work we consider the dependence of solutions to a partial differential equations system on its data. The problem of interest is a coupled model of nonlinear flow an transport in porous media, with applications, e.g. to environmental modeling. The model of flow we consider is known as the non-Darcy model, and its solutions: the velocity, and pressure unknowns, depend on the coefficients of permeability and inertia, and other data such as boundary conditions. In turn, the transport solutions depend on the velocity of the fluid, and on boundary and initial conditions. Furthermore, one can be interested in a particular quantity computable from the flow and transport solutions, and represented by a functional. In this work we evaluate rigorously the sensitivity, i.e., the derivative, of the solutions, or of the quantity of interest, upon the data.
Due to its delicate nature, the sensitivity is evaluated either in a direct way, called Forward Sensitivity, or via an adjoint method, which only uses a variational form. Our first contribution is that we find a way to find the sensitivity for the coupled flow and transport model without having to solve multiple flow problems. Second, we prove the well-posedness of the flow problem, and set up the numerical approximation using the framework similar to that of expanded mixed finite element methods.
Next, the numerical approximation of the problem leads to a nonlinear system of discrete equations, which is difficult to solve. To aid in solving this system, we propose to take advantage of sensitivity analysis, which is used in a novel way within a homotopy framework. The theoretical results in this thesis are illustrated with numerical simulations. The code in Python for the examples is provided.
Ken's major professor is Prof. Malgo Peszynska.