Novel Virtual Element Methods with Applications in Interface Problems

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Interface problems arise from many important complex multiphysics and biological systems, such as those involving the evolutions of multi-fluid/material interfaces, tumor growth, or stem cell deformation. Computer-aided simulation is a cost-friendly tool for the studies of these challenging interface problems. To approximate the governing mathematical equations of these systems, Virtual element method (VEM) is an emerging powerful tool in the scientific computing community. The objective of this research project is to develop both theoretical and practical aspects of various VEMs. Being able to reliably answer the question "can we trust our simulation results?" justifies the use of VEMs in simulating these complex systems. Meanwhile, this project strives to provide the public with a state-of-the-art VEM computer program that saves valuable computing resources. In addition, this research project creates opportunities to pass the torch on to graduate students to become the next generation computational mathematicians.Solving elliptic partial differential equations with high-contrast diffusion coefficients play a central role in the modeling these complex systems. This project shall develop an in-depth robust a priori error analysis of VEM on elliptic interface problems. Different from the existing VEM analysis, this project devises a new novel paradigm to study the error analysis for the interface VEM, and further clarifies the dependence of the VEM convergence on the polytopal mesh geometries, justifying the VEM's applicability on interface-fitted mesh which may become extremely irregular or degenerate near the interfaces. Meanwhile, this project learns from the novelty of VEM framework to improve the analyses of traditional approaches for interface problems. The VEM's meta-formulation for elliptic problems enables us to construct immersed finite element spaces naturally in higher order and/or in 3-D. Higher order interface VEMs, the a posteriori error estimation, and the adaptive polytopal mesh refinement are to be studied to render VEM more efficient and effective. This integrated study enables us to attack the challenging 3-D interface problems, which, in turn, broadens the scope in terms of both theory and tools for the whole numerical partial differential equation community. Last but not least, a portable and highly-vectorized VEM software library shall be made publicly available, including the semi-structured interface-fitted mesh generation, vectorized assembling, polytopal adaptivity, and fast multigrid solvers. The portability of the computer code enables the researchers to incorporate the VEM into existing software libraries dealing with interface problems, thus facilitating the interdisciplinary research in simulating those complex systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.