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An -estimate of the finite element error is proved for a Dirichlet and a Neumann boundary value problem on a three-dimensional, prismatic and non-convex domain that is discretized by an anisotropic tetrahedral mesh. To this end, an approximation error estimate for an interpolation operator that is preserving the Dirichlet boundary conditions is given. The challenge for the Neumann problem is the proof of a local interpolation error estimate for functions from a weighted Sobolev space.
Dick proved that all dyadic order 2 digital nets satisfy optimal upper bounds on the -discrepancy. We prove this for arbitrary prime base b with an alternative technique using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds on the discrepancy function in Besov spaces with dominating mixed smoothness for a certain parameter range, and enlarge that range for order 2 digital nets. The discrepancy function in Triebel-Lizorkin and Sobolev spaces with dominating mixed...
We prove stability and derive error estimates for the recently introduced central discontinuous Galerkin method, in the context of linear hyperbolic equations with possibly discontinuous solutions. A comparison between the central
discontinuous Galerkin method and the regular discontinuous
Galerkin method in this context is also made.
Numerical experiments are provided to validate the quantitative
conclusions from the analysis.
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