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This paper reviews the variational multiscale stabilization of standard finite element methods
for linear partial differential equations that exhibit multiscale features. The stabilization is of …

Abstract Discontinuous Petrov–Galerkin (DPG) methods are made easily implementable
using “broken” test spaces, ie, spaces of functions with no continuity constraints across mesh …

We propose and analyze a discontinuous Petrov--Galerkin (DPG) method for convection-
dominated diffusion problems that provides robust L^2 error estimates for the field variables …

We introduce a new Petrov-Galerkin multiscale method for the numerical approximation of
the Helmholtz equation with large wave number $\kappa $ in bounded domains in $\mathbb …

An absolutely stable weak Galerkin (WG) finite element method is introduced and analysed
for the Helmholtz equation. This means that the stability and well-posedness of the method …

In this paper, which is the second in a series of two, the preasymptotic error analysis of the
continuous interior penalty finite element method (CIP-FEM) and the FEM for the Helmholtz …

A spacetime discontinuous Petrov--Galerkin (DPG) method for the linear time-dependent
Schrödinger equation is proposed. The spacetime approach is particularly attractive for …

There is tremendous potential in using neural networks to optimize numerical methods. In
this paper, we introduce and analyze a framework for the neural optimization of discrete …

A preasymptotic error analysis of the finite element method (FEM) and some continuous
interior penalty finite element method (CIP-FEM) for the Helmholtz equation in two and three …