scholarly journals Discrete Energy Laws for the First-Order System Least-Squares Finite-Element Approach

2018 ◽  
pp. 3-20
Author(s):  
J. H. Adler ◽  
I. Lashuk ◽  
S. P. MacLachlan ◽  
L. T. Zikatanov
Keyword(s):  
Finite Element ◽  
Least Squares ◽  
Order System ◽  
Discrete Energy ◽  
Finite Element Approach ◽  
First Order ◽  
Element Approach

Hybrid First-Order System Least Squares Finite Element Methods with Application to Stokes Equations

2013 ◽  
Vol 51 (4) ◽  
pp. 2214-2237 ◽  
Author(s):  
K. Liu ◽  
T. A. Manteuffel ◽  
S. F. McCormick ◽  
J. W. Ruge ◽  
L. Tang
Keyword(s):  
Finite Element ◽  
Least Squares ◽  
Finite Element Methods ◽  
Stokes Equations ◽  
Order System ◽  
First Order

scholarly journals Finite Element Approach for the Solution of First-Order Differential Equations

2020 ◽  
Vol 08 (10) ◽  
pp. 2072-2090
Author(s):  
André Schmidt ◽  
Horst R. Beyer ◽  
Matthias Hinze ◽  
Evangelos N. Vandoros
Keyword(s):  
Finite Element ◽  
Differential Equations ◽  
Finite Element Approach ◽  
First Order ◽  
Element Approach ◽  
First Order Differential Equations

A Newton Div-Curl Least-Squares Finite Element Method for the Elliptic Monge–Ampère Equation

2019 ◽  
Vol 19 (3) ◽  
pp. 631-643 ◽  
Author(s):  
Chad R. Westphal
Keyword(s):  
Finite Element ◽  
Least Squares ◽  
Convergence Rates ◽  
Order System ◽  
Classical Solutions ◽  
Convex Domains ◽  
First Order ◽  
Convergence Results ◽  
Bounded Data ◽  
Monge Ampere

AbstractThis paper develops a new finite element approach for the efficient approximation of classical solutions of the elliptic Monge–Ampère equation. We use an outer Newton-like linearization and a first-order system least-squares reformulation at the continuous level to define a sequence of first-order div-curl systems. For problems on convex domains with smooth and appropriately bounded data, this framework gives robust results: convergence of the nonlinear iteration in a small number of steps, and optimal finite element convergence rates with respect to the meshsize. Numerical results using standard piecewise quadratic or cubic elements for all unknowns illustrate convergence results.

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