A posteriori error estimates for the virtual element method

An a posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Upper and lower bounds of the error estimator with respect to the VEM approximation error are proven. The error estimator is used to drive adaptive mesh refinement in a number of test problems. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing

Analysis of Discontinuous Galerkin Methods and Applications to Kinetic Equations

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Uniformly convergent iterative methods for discontinuous Galerkin discretizations

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Discontinuous Galerkin methods for the multi-dimensional Vlasov-Poisson problem

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L2-estimates for the DG IIPG-0 scheme

We discuss the optimality in L^2 of a variant of the Incomplete Discontinuous Galerkin Interior Penalty method (IIPG) for second order linear elliptic problems. We prove optimal estimate, in two and three dimensions, for the lowest order case under suitable regularity assumptions on the data and on the mesh. We also provide numerical evidence, in one dimension, of the necessity of the regularity assumptions

Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients

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Substructuring preconditioners for an {hh}-{pp} domain decomposition method with interior penalty mortaring

We propose and study an iterative substructuring method for an $$h$$h-$$p$$p Nitsche-type discretization, following the original approach introduced in Bramble et al. Math. Comp. 47(175):103–134, (1986) for conforming methods. We prove quasi-optimality with respect to the mesh size and the polynomial degree for the proposed preconditioner. Numerical experiments assess the performance of the preconditioner and verify the theory

A block solver for the exponentially fitted IIPG--0 method

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