Abstract
In this paper, the local multilevel methods for discontinuous Galerkin finite element on adaptively refined meshes are considered. By the abstract Schwarz theory, we analyze the convergence rate of the proposed algorithms for smooth and highly discontinuous coefficients separately. It is shown that in the case of smooth coefficients, the convergence rate of the local multilevel methods is independent of mesh sizes and mesh levels. If the coefficients have large jumps, the algorithms are sub-optimal, i.e., the convergence rate is only dependent on mesh levels.
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