Abstract. More than a decade ago, Bramble, Pasciak and Xu developed a framework in analyzing the multigrid methods with nonnested spaces or noninherited quadratic forms. It was subsequently known as the BPX multigrid framework, which was widely used in the analysis of multigrid and domain decomposition methods. However, the framework has an apparent limit in the analysis of nonnested V-cycle methods, and it produces a variable V-cycle, or nonuniform convergence rate V-cycle methods, or other nonoptimal results in analysis thus far.This paper completes a long-time effort in extending the BPX multigrid framework so that it truly covers the nonnested V-cycle. We will apply the extended BPX framework to the analysis of many V-cycle nonnested multigrid methods. Some of them were proven previously only for two-level and W-cycle iterations. Some numerical results are presented to support the theoretical analysis of this paper.

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To deal with the divergence-free constraint in a double curl problem: curl µ −1 curl u = f and div εu = 0 in Ω, where µ and ε represent the physical properties of the materials occupying Ω, we develop a δ-regularization method: curl µ −1 curl u δ + δεu δ = f to completely ignore the divergence-free constraint div εu = 0. It is shown that u δ converges to u in H(curl ; Ω) norm as δ → 0. The edge finite element method is then analyzed for solving u δ. With the finite element solution u δ,h , quasi-optimal error bound in H(curl ; Ω) norm is obtained between u and u δ,h , including a uniform (with respect to δ) stability of u δ,h in H(curl ; Ω) norm. All the theoretical analysis is done in a general setting, where µ and ε may be discontinuous, anisotropic and inhomogeneous, and the solution may have a very low piecewise regularity on each material subdomain Ω j with u, curl u ∈ (H r (Ω j)) 3 for some 0 < r < 1, where r may be not greater than 1/2. To establish the uniform stability and the error bound for r ≤ 1/2, we have respectively developed a new theory for the K h ellipticity (related to mixed methods) and a new theory for the Fortin interpolation operator. Numerical results presented confirm the theory.

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