The energy-conserving sampling and weighting (ECSW) method is a hyper-reduction originally developed for accelerating the performance of Galerkin projection-based reduced-order models (PROMs) associated with large-scale finite element models, when underlying projected operators need to be frequently recomputed as in parametric and/or nonlinear problems. In this paper, extended Petrov–Galerkin P...

This article is concerned with developing efficient discontinuous Galerkin methods for approximating viscosity (and classical) solutions of fully nonlinear second-order elliptic and parabolic partial differential equations (PDEs) including the Monge–Ampère equation and the Hamilton–Jacobi–Bellman equation. A general framework for constructing interior penalty discontinuous Galerkin (IP-DG) meth...

The goal of this work is to derive and justify a multilevel preconditioner for symmetric discontinuous approximations of second order elliptic problems. Our approach is based on the following simple idea. The finite element space V of piece-wise polynomials of certain degree that are discontinuous on the partition T0 is projected onto the space of piece-wise constants on the same partition. Thi...

Complementarity problems and variational inequalities arise in a wide variety of areas, including machine learning, planning, game theory, and physical simulation. In all of these areas, to handle large-scale problem instances, we need fast approximate solution methods. One promising idea is Galerkin approximation, in which we search for the best answer within the span of a given set of basis f...

We study the problem -? u = f , where has a point-singularity. In particular, we are interested in ? x 0 Dirac delta with support but singularities of form ~ | ? ?s also considered. prove stability Galerkin projection on graded meshes weighted spaces, weights given by powers distance to . recover optimal rates convergence for finite element method these meshes. Our approach is general and holds...

In this paper we present an efficient discretization method for the solution of the unsteady incompressible Navier-Stokes equations based on a high order (Hybrid) Discontinuous Galerkin formulation. The crucial component for the efficiency of the discretization method is the disctinction between stiff linear parts and less stiff non-linear parts with respect to their temporal and spatial treatm...

A new approach to flux limiting for systems of conservation laws is presented. The Galerkin finite element discretization / L2 projection is equipped with a failsafe mechanism that prevents the birth and growth of spurious local extrema. Within the framework of a synchronized flux-corrected transport (FCT) algorithm, the velocity and pressure fields are constrained using node-by-node transforma...

We consider finite element methods for the Darcy equations that are designed to work with standard, low order C finite element spaces. Such spaces remain a popular choice in the engineering practice because they offer the convenience of simple and uniform data structures and reasonable accuracy. A consistently stabilized method [20] and a least-squares formulation [18] are compared with two new...

It is well known that the classical local projection method as well as residual-based stabilization techniques, as for instance streamline upwind Petrov-Galerkin (SUPG), are optimal on isotropic meshes. Here we extend the local projection stabilization for the Navier-Stokes system to anisotropic quadrilateral meshes in two spatial dimensions. We describe the new method and prove an a priori err...

In this paper we propose a state estimation method for linear parabolic partial differential equations (PDE) that accounts for errors in the model, truncation, and observations. It is based on an extension of the Galerkin projection method. The extended method models projection coefficients, representing the state of the PDE in some basis, by means of a differential-algebraic equation (DAE). Th...