A least squares method for mixed variational formulations of partial differential equations

Bacuta, ConstantinWe present a general framework for solving mixed variational formulations of partial differential equations. The method relates the theories of least squares finite element methods, approximating solutions to elliptic boundary value problems, and approximating solutions to symmetric saddle point problems. A general preconditioning strategy for the proposed framework is also given that utilizes the theory of multilevel preconditioners. One of the main advantages of the method is that an inf-sup condition is automatically satisfied at the discrete level for standard choices of test and trial spaces. Another benefit is that the method allows for the use of nonconforming trial spaces. In addition, the framework allows the freedom to choose the inner product on the test space, which is useful when solving PDEs, or first order systems of PDEs, with parameters and/or discontinuous coefficients. The proposed iterative solver does not require explicit bases for the trial spaces as well. Applications of the method to second order elliptic interface problems, reaction diffusion equations, and time-harmonic Maxwell's equations are presented. Numerical results in 2D and 3D, for both convex and non-convex domains, are given to support the methodology, including problems with discontinuous or highly oscillatory coefficients, low regularity of the solution, and boundary layers.University of Delaware, Department of Mathematical SciencesPh.D