Least-Squares Finite Element Methods

By Pavel B. Bochev, Max D. Gunzburger. Applied Mathematical Sciences, 2009

476 citations recorded by the source corpus.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/375cb6153d898bf1287fde06740c62a88d51691b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Electrical and Electronic Engineering

In Mechanics of Materials

Citations in the corpus

An augmented velocity–vorticity–pressure formulation for the Brinkman equations (2015, background): “The so-called augmented mixed finite elements (also known as Galerkin least-squares methods [12, 20, 21]) can be regarded as a stabilization technique where some terms are added to the variational formulation so that, either the resulting augmented variational formulations are defined by strongly co”
Element differential method and its application in thermal‐mechanical problems (2018, unknown): “Dijkl x ð Þ ∂ (2)uk x ð Þ ∂xl∂xj þ ∂Dijkl x ð Þ ∂xj ∂uk x ð Þ ∂xl − ∂ ~k x ð ÞT x ð Þ ∂xi þ bi x ð Þ 1⁄4 0; x ∈ Ω; (6)”
A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations (2018, methodology): “In the ﬁrst approach, similar in spirit to the Least-Square Finite Element Methods (LSFEM) [26], we minimize the squared residual over the entire computational domain, with the random PDE solution represented in form of a feed-forward fully-connected deep residual network.”
Efficient training of physics‐informed neural networks via importance sampling (2021, methodology): “in which λ1 and λ2 are weight parameters, analogous to collocation finite element method in which weights are used to adjust the relative importance of each residual term [43].”
A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two (2013, methodology): “When R > √ 2, the exact solution satisfies ψ ∈ C∞ ( Ω ) , while ψ ∈W (Ω), p ∈ [1, 4), when R = √ 2.”
Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, background): “This variational formulation for non-self adjoint elliptic operators (as well as similar variational principles for parabolic equations, gradient flows and other problems) have been discovered and rediscovered multiple times, in the PDE community (see [28, 82]), by convex analysts [54, 86] and by nu”