Conservation and efficiency in least squares finite element methods

finite elements;least-squares finite element methods;multigrid;high-order;discontinuous least-squares;Navier-Stokes;Stokes;edge elements;H(curl) multigrid

Lai, James

关键词: finite elements;    least-squares finite element methods;    multigrid;    high-order;    discontinuous least-squares;    Navier-Stokes;    Stokes;    edge elements;    H(curl) multigrid;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/34237/Lai_James.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

Two of the main aspects in the numerical solution of partial differential equations include accurate discretizations and efficient solutions of the algebraic equations.With respect to discretizations, conservation is often sought after.However, least-squares finite element methods are known to be not mass conserving when solving fluid flow problems. In this dissertation we develop mass conservative least-squares finite element methods for the Stokes and Navier-Stokes equations through the use of discontinuous finite element spaces.We formulate two divergence free formulations using both a discontinuous stream-function and a locally divergence free basis and we present a thorough numerical study of both methods.This dissertation is also concerned with the efficient solution of algebraic equations via multigrid methods.Specifically, we formulate multigrid methods for high-order H(curl) conforming finite elements.Such elements are often used in mimetic discretizations of Maxwell's equations often solved in electromagnetic applications.Efficient multigrid methods for high-order H^1 conforming finite elements and also for the lowest-order H(curl) basis have been extensively studied in recent research.We draw upon elements of both algorithms to formulate multigrid methods for high-order H(curl) finite elements for hierarchical and interpolatory type.