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

## Palavras-chave:

Preconditioning, Iterative methods, Finite Element Method, Mixed approximations## 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.