Abstract Mathematics lies at the heart of engineering science and is very important for
capturing and modeling of diverse processes. These processes may be naturally-occurring …

In this work, we formulate and analyze a geometric multigrid method for the iterative solution
of the discrete systems arising from the finite element discretization of symmetric second …

We consider scalar semilinear elliptic PDEs where the nonlinearity is strongly monotone, but
only locally Lipschitz continuous. We formulate an adaptive iterative linearized finite element …

We consider a general nonsymmetric second-order linear elliptic partial differential equation
in the framework of the Lax–Milgram lemma. We formulate and analyze an adaptive finite …

The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to
compute an approximation of user-prescribed accuracy at quasi-minimal computational …

We review different techniques to enforce essential boundary conditions, such as the
(nonhomogeneous) Dirichlet boundary condition, within a discrete variational framework …

We analyze a goal-oriented adaptive algorithm that aims to efficiently compute the quantity
of interest G (u⋆) with a linear goal functional G and the solution u⋆ to a general second …

In this work, we formulate and analyze a geometric multigrid method for the iterative solution
of the discrete systems arising from the finite element discretization of symmetric second …

A MATLAB implementation of hierarchical shape functions on 2D rectangles is explained
and available for download. Global shape functions are ordered for a given polynomial …

This chapter provides an overview of state-of-the-art adaptive finite element methods
(AFEMs) for the numerical solution of second-order elliptic partial differential equations …