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Prof. Howell's research in numerical analysis and computational mathematics is centered around several different aspects of numerical approximation of PDEs.  Prof. Howell is  interested in the theoretical underpinnings of the finite element method, in particular compatibility conditions for mixed methods.  He is also interested in the development of finite element methods that directly approximate quantities of significant physical interest, such as fluid stresses, as well as those that relax regularity requirements.  These methods are often employed for continuum models in fluid and structure dynamics.  Finally, he is interested in fast and efficient linear solvers and decoupling approaches for time-dependent problems.  

Prof. Howell is also interested in applications of math in the sciences, and has worked with chemists and physicists on projects ranging from solubility parameters to nanostructures.

Prof. Howell is very interested in undergraduate research in Mathematical Sciences, and mentors several students in research projects each summer.  The projects range from fluid-structure interaction problems to numerical linear algebra to applications of differential equations.

• Current Activities: Undergraduate Research in the Mathematical Sciences; Finite Element Methods for Fluids and Structures; Applications of Differential Equations in the Natural and Social Sciences; Direct Solution Methods for Large Sparse Linear Systems; Numerical and Computational Analysis of Arterial Blood Flow; Numerical Methods for Coupled Multiscale Problems in Fluid/Fluid and Fluid/Structure Interaction.

• General Interests: Numerical and Computational Analysis; Numerical Solution of Partial Differential Equations; Computational Fluid Dynamics; Finite Element Methods; Saddle Point Problems; Inf-Sup Conditions; Temporal Integration Methods for Systems of Ordinary Differential Equations; Operator-Splitting Methods; Defect Correction Methods; Continuation Methods; Newtonian and Non-Newtonian Fluid Flow; Reaction-Diffusion Equations; Flow in Porous Media; Iterative Linear and Nonlinear Solvers.