On splitting schemes of predictor-corrector type in mixed finite element method

In this work we develop a previously proposed approach to constructing vector splitting schemes for heat transfer problem solved by mixed finite element method on rectangular meshes. As was shown numerically before, a particular flux splitting scheme based the alternating direction scheme for flux divergence has no convergence for some smooth test solutions. We provide theoretical analysis of the stability estimates for the scheme based on the eigensystem information. The main drawback of that particular flux splitting scheme is the nonzero component of the heat flux in the kernel of divergence operator.
Based on the analysis and numerical experiments we suggest, explain and verify numerically that flux splitting schemes obtained from predictor-corrector schemes for flux divergence don’t have this drawback. The main conclusion is that due to the presence of simple and strong stability estimates one should prefer using predictor-corrector type of schemes for the heat flux rather than others.