：Prof. Shuyu Sun
Speaker：Prof. Shuyu Sun
King Abdullah University of Science Technology
Title: Multiscale Finite Elements: From the Solution of Best Approximation to the Subspace of Best Approximation
Time：21 July 2019, 16:00
Abstract: Multiscale processes are universal phenomena, multiscale modeling simulation are required for many important applications. For example, multiscale modeling simulation of flow transport in the Earth's subsurface is required to make decisions associated with the management of subsurface reservoirs, with applications in groundwater contamination, carbon sequestration, petroleum exploration recovery, many others. In this talk, we discuss multiscale finite element approximation, which is an important numerical method for efficient computation of subsurface simulations. For better illustration, we consider single-phase flow in porous media as our model problem, which is a second-order elliptic partial differential equation, even though extension to more complicated physics is possible. We will first review finite different methods finite element methods as well as their comparison; in particular, we will demonstrate the advantage of finite element methods from both theoretical viewpoint practical viewpoint. We will recall Céa's lemma then quickly show that the finite element solution is the best approximation to the full-space solution (i.e. the exact solution) among all possible functions within the finite element space in respect to a certain well-defined energy norm. We will then talk about how to improve the choice of the finite element space to reflect multiscale phenomena. We formulate the problem mathematically as follows: we seek the subspace of best approximation, which is the subspace of a certain given large reference space with least approximation in term of approximating the exact solution. From it, we then introduce the Proper Orthogonal Decomposition (POD) for model reduction Generalized Multiscale Finite Element Methods (GMsFEM) for porous media flow. Finally, we present our recent work on a new novel mixed GMsFEM for porous media flow, with surprisingly good numerical examples.