New mathematical and computational tools for accurately modeling physical phenomena that exhibit widely varying behavior at different
spatial and temporal scales is currently of great interest in nearly every area of modern science and engineering. As a result, the study
and development of new reliable analytical and numerical techniques for modeling such phenomena, often referred to as "multiscale modeling" techniques, are currently among the most important and vibrant research areas in applied and computational mathematics.
In this sequence of two tutorial lectures on numerical methods, we will give an overview of some background material in numerical methods for
partial differential equations (PDE); this material will help provide some of the mathematical and numerical tools for getting the most out of the workshop talks on multiscale modeling in the weeks to follow.
In the first lecture, we will give an overview of standard discretization techniques for various types of PDE, and will then focus primarily on
finite element-type discretizations for nonlinear elliptic and parabolic equations. We will go over basic a priori error estimates to understand the convergence of these methods, and will then derive some a posteriori error estimates for using adaptive techniques for multiscale modeling. We describe a complete adaptive multilevel finite element modeling
algorithm and its implementation (FEtk, the Finite Element ToolKit),and then discuss its properties, including aspects of the various
components such as linear and nonlinear solvers.
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