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Numerical Methods for PDEs: In Occasion of Raytcho Lazarov's 70th Birthday

Panayot Vassilevski, Lawrence Livermore National Laboratory
Improving Conservation Properties of First-Order System Least Squares Finite-Element Methods


The first-order system least-squares (FOSLS) finite element method for solving partial differential equations has many advantages, including the construction of symmetric positive definite algebraic linear systems that can be solved efficiently with multilevel iterative solvers. However, one drawback of the method is the potential lack of conservation of certain properties. One such property is conservation of mass. In this talk we describe a strategy for achieving mass conservation for a FOSLS system by changing the minimization process to that of a constrained minimization problem. If the space of corresponding Lagrange multipliers contains the piecewise constants, then local mass conservation is achieved similarly to the standard mixed finite element method. To make the strategy more robust and not add too much computational overhead to solving the resulting saddle-point system an overlapping Schwarz process is applied which we illustrate with numerical tests.

This talk is based on the report: J.H. Adler, P.S. Vassilevski, Improving conservation for first-order system least squares finite-element methods. Lawrence Livermore National Laboratory Technical Report LLNL-PROC-579552, .

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344.