A low-order preconditioner for high-order element-wise divergence con- stant finite element spaces

Resumo

Mixed finite element problems are a class of problems that arises when modeling several physical
phenomena, such as in computational fluid dynamics, structural analysis, optimization, etc. Designing efficient
iterative schemes for such a family of approximations has been the subject of several works in the past decades.
However, its success is intimately related to the proper definition of a preconditioner, i. e., the projection of the
original algebraic system to an equivalent one with better spectral properties. In recent work, we have proposed
a new class of H(div)-conforming finite element spaces with element-wise constant divergent. This family of
elements was designed to improve reservoir simulation computational cost and are obtained by choosing the lower
order space with piece-wise constant normal fluxes incremented with divergence-free higher-order functions. In

this work, we propose an iterative scheme to solve problems arising in the context of the above mentioned element-
wise constant divergence approximation spaces. The strategy consists on using the matrix of linear fluxes as a

preconditioner to solve the higher-order flux problem. The latter is solved iteratively by means of a conjugate
gradient scheme. In the presented numerical tests, this strategy has shown to be convergent in a few iterations for
different problems in 2D and 3D. In addition, as internal fluxes are condensed, only boundary variables need to
be computed. This strategy relates to the MHM technique and can be efficiently used to access fast multi-scale
approximations in future work.