We outline a solution method for mixed finite element discretizations based on dissecting the problem into three separate steps. The first handles the inhomogeneous constraint, the second solves the flux variable from the homogeneous problem, whereas the third step, adjoint to the first, finally gives the Lagrangian multiplier. We concentrate on aspects involved in the first and third step mainly, and advertise a multi-level method that allows for a stable computation of the intermediate and final quantities in optimal computational complexity.
In cases of human malaria, children suffer very high rates of morbidity and mortality. To analyze the mechanisms involved in age-dependent protection against malaria, we investigated the characterization of immune responses to Plasmodium yoelii 17XNL (P.y 17XNL) in young (3 weeks) and middle-aged (8 months) C57BL/6 mice. In this study, we found that 100% of young mice succumbed to P.y 17XNL infection with higher parasitemia, while middle-aged mice were able to clear blood parasites and no mortality was observed. These observations suggested that the young C57BL/6 mice were susceptible to P.y 17XNL infection, whereas the middle-aged mice were resistant. Cellular analysis revealed that both the numbers of splenic myeloid dendritic cells (mDCs) as well as the expression of DC maturation markers were higher in middle-aged mice than those in young mice. The numbers of IgG1- or IgG2a-secreting B cells increased markedly in middle-aged mice after infection with P.y 17XNL. The dynamic change of the number of CD4+CD25+Foxp3+ regulatory T cells (Tregs) in mice infected with P.y 17XNL was also different between the two groups. In addition, the levels of IFN-γ and NO increased in both groups during early parasite infection, while there was also an obvious increase in IL-4 production in the infected middle-aged mice. The change in IL-10 levels following infection was consistent with that of the change in the number of Tregs. The survival of middle-aged mice following P.y 17XNL infection was dependent upon the establishment of effective Th1 and Th2 responses and a successful switch between Th1 and Th2 responses, as well as appropriate functioning of Tregs.