Isoparametric multigrid method for reaction–diffusion equations on two-dimensional domains

The reaction–diffusion equation on curved domains Ω is considered. The curved boundary is approximated by using isoparametric finite elements. To be able to apply multigrid methods a sequence of finite element triangulations is constructed, which gives a sequence of domains Ωk, k=0,1,…,ℓ, approximating the domain Ω. In the case of problems on domains with nonpolynomial boundaries the corresponding finite element spaces are usually nonnested. Therefore, we have to consider solution methods with nonnested spaces. We define a bijection from one approximating domain to another. On this basis a new intergrid transfer operator is constructed and its stability is proved. A pure isoparametric approach is used for obtaining a nonnested multigrid method. An optimal convergence order in an energy norm for the two-level algorithm is proved.