This book introduces the basic ideas to build discontinuous Galerkin methods and, at the
same time, incorporates several recent mathematical developments. The presentation is to a …

Originally introduced in [146, 158], Hybrid High-Order (HHO) methods provide a framework
for the discretisation of models based on Partial Differential Equations (PDEs) with features …

Self-adaptive discretization methods are now an indispensable tool for the numerical
solution of partial differential equations that arise from physical and technical applications …

This paper introduces a quasi-interpolation operator for scalar-and vector-valued finite
element spaces constructed on affine, shape-regular meshes with some continuity across …

We present equilibrated flux a posteriori error estimates in a unified setting for conforming,
nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two …

Finite element discretizations of problems in computational physics often rely on adaptive
mesh refinement (AMR) to preferentially resolve regions containing important features …

This article is a review of basic concepts and tools devoted to a posteriori error estimation for
problems solved with the finite element method. For the sake of simplicity and clarity, we …

We present new results in the numerical analysis of singularly perturbed convection‐
diffusion‐reaction problems that have appeared in the last five years. Mainly discussing …

In this contribution we consider localized, robust, and efficient a posteriori error estimation of
the localized reduced basis multiscale (LRBMS) method for parametric elliptic problems with …

We derive a posteriori error estimates for the discretization of the heat equation in a unified
and fully discrete setting comprising the discontinuous Galerkin, various finite volume, and …