Hybridizing Finite Element Methods for Second-Order Elliptic Problems in High Dimension

Recently the idea of combining FDMs (the Finite Difference Methods) and FEMs (the Finite Element Methods) has been successfully applied to solve several interesting problems, including elliptic problems and the Navie-Stokes equations. This idea is useful to solve high dimensional PDEs. In this talk, we briefly review the recent progress on the hybridization of FDMs and FEMs to solve elliptic equations. We present stability and error analyses. We will focus on some norm equivalence results between continuous and discrete norms in arbitrary finite-dimensional Q1-finite element spaces, which arising from the above hybridization. Also, some numerical results will be presented. This is a joint work with Xinlong Feng(XJU), Yinnian He(XJTU), Mapeng Ma(CUPK), Chenhong Shi(XJU). Dongwoo Sheen received BA and MA at SNU in 1981 and 1983, and his PhD under the guidance of Prof. Jim Douglas, Jr. at Purdue University in 1991. Then he went to Pavia, Italy as a CNR postdoctoral fellow under Franco Brezzi’s guidance. He then went back to Purdue University as a postdoc. Since 1993, he worked for SNU until 2023. Currently he is a Professor Emeritus at SNU and a Distinguished Professor at Xinjiang University. He held visiting professorship at the New South Wales University(1999), Purdue University(2006–2007), Texas A&M University(2013–2014), Kyoto University(2015), and the Ocean University of China (2023). His research interests include Numerical Analysis and Scientific Computation in several application areas including fluid and solid mechanics, electrodynamics, math finance, and math biology. Specifically he has contributed in developing several fundamental Nonconforming Finite Element Methods and parallel algorithms based on Laplace Transform Methods. He served as a President for Applied Math Forum, the East Asia Section of SIAM, the Korean Society for Math Biology, and the Korean Society of Computational Sciences.