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Iterated numerical homogenization for multi-scale elliptic equations with monotone nonlinearity [article]

Xinliang Liu, Eric Chung, Lei Zhang
2021 arXiv   pre-print
In this paper, we study the numerical homogenization for multi-scale elliptic PDEs with monotone nonlinearity, in particular the Leray-Lions problem (a prototypical example is the p-Laplacian equation)  ...  We develop the iterated numerical homogenization scheme by combining numerical homogenization methods for linear equations, and the so-called "quasi-norm" based iterative approach for monotone nonlinear  ...  In this paper we propose the iterated numerical homogenization method for multi-scale elliptic equations with monotone nonlinearity.  ... 
arXiv:2101.00818v1 fatcat:wo65bsih2zfi5g3by76xpyoqaq

On the Implementation of a Heterogeneous Multi-scale Finite Element Method for Nonlinear Elliptic Problems [chapter]

Patrick Henning, Mario Ohlberger
2012 Advances in DUNE  
In this contribution, we formulate a heterogeneous multiscale finite element method (HMM) for monotone elliptic operators.  ...  The algorithm is validated by numerical experiments and can be used to effectively determine homogenized solutions.  ...  In the following, we are concerned with solving the following elliptic multiscale problem: Problem 0.1 (Nonlinear elliptic equation with fast oscillations).  ... 
doi:10.1007/978-3-642-28589-9_11 fatcat:pdx7uocnyfa7ra3qr6tpz2af5u

Page 5743 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews  
[Xu, Xue Jun] (PRC-ASBJ-CP; Beijing) Cascadic multigrid method for elliptic problems. (English summary) East-West J. Numer. Math. 7 (1999), no. 3, 199-209.  ...  (English summary) Numer. Methods Partial Differential Equations 16 (2000), no. 1, 1-10.  ... 

An Accelerated Method for Nonlinear Elliptic PDE

Hayden Schaeffer, Thomas Y. Hou
2016 Journal of Scientific Computing  
We propose two numerical methods for accelerating the convergence of the standard fixed point method associated with a nonlinear and/or degenerate elliptic partial differential equation.  ...  Numerical examples are shown for Bellman's equation, Isaacs' equation, Pucci's equations, the Monge-Ampère equation, a variant of the infinity Laplacian, and a system of nonlinear equations.  ...  For example, in [25] , monotone stencils for fully nonlinear elliptic equations of the form H (D 2 u) = 0 are constructed which have algebraic rates of convergence.  ... 
doi:10.1007/s10915-016-0215-8 fatcat:hr4xblhcbnggxnd2t46hruvqhu

A multi-scale DNN algorithm for nonlinear elliptic equations with multiple scales [article]

Xi-An Li, Zhi-Qin John Xu, Lei Zhang
2020 arXiv   pre-print
In this work, we utilize the multi-scale DNN-based algorithm (MscaleDNN) proposed by Liu, Cai and Xu (2020) to solve multi-scale elliptic problems with possible nonlinearity, for example, the p-Laplacian  ...  Several numerical examples of multi-scale elliptic problems with separable or non-separable scales in low-dimensional and high-dimensional Euclidean spaces are used to demonstrate the effectiveness and  ...  In comparison, numerical homogenization methods, can solve linear multi-scale problems [14, 24, 35] and nonlinear multi-scale problems [1, 9] on the coarse scale, without resolving all the fine scales  ... 
arXiv:2009.14597v2 fatcat:n4b737lyoveedfkmj5yxgcl6om

subject index volumes 181-190

2006 Journal of Computational and Applied Mathematics  
of special functions 190 270 Numerical integration 184 475; 187 41; 189 341 Numerical methods for DAEs 189 34 Numerical methods for ODEs 189 34 Numerical methods for stochastic equations 185 422  ...  190 304 Multi-index Hermite polynomials 182 165 Multiple scales 190 3 Multiple solution 181 467; 189 412 Multiple-scale method 190 22 Multiplicative noise 184 404 Multiquadric 188 265 Multi-rate  ... 
doi:10.1016/s0377-0427(06)00062-8 fatcat:cpf5lksg7rgzzkz25xgdhtvu64

Subject index to volumes 71–80

1997 Journal of Computational and Applied Mathematics  
curvature Monotone matrix Monotone operators Monotonicity Monte Carlo simulation MSOR method Multi-level basis Multi-step methods Multigrid M ultigrid method Multilevel method Multiple integrals  ...  Electric field Electrohydrodynamic stability Electromagnetic loss Elliptic equations Elliptic problem Elliptic problems Error bound for asymptotic expansion Error control Error estimates  ... 
doi:10.1016/s0377-0427(97)80101-x fatcat:o46ccj4sq5gblfrhk57b6acvtu

Adaptive finite difference methods for nonlinear elliptic and parabolic partial differential equations with free boundaries [article]

Adam M. Oberman, Ian Zwiers
2015 arXiv   pre-print
Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic Partial Differential Equations (PDEs).  ...  In this article we combine monotone finite difference methods with an adaptive grid refinement technique to produce a PDE discretization and solver which is applied to a broad class of equations, in curved  ...  Introduction In this article we numerically approximate a class of nonlinear elliptic and parabolic PDEs using monotone finite difference methods.  ... 
arXiv:1412.3057v2 fatcat:fddpjwrl4jdalfop2smxaq57x4

Finite difference methods for the Infinity Laplace and p-Laplace equations [article]

Adam M. Oberman
2012 arXiv   pre-print
We build a semi-implicit solver, which solves the Laplace equation as each step. It is fast in the sense that the number of iterations is independent of the problem size.  ...  We build convergent discretizations and semi-implicit solvers for the Infinity Laplacian and the game theoretical p-Laplacian. The discretizations simplify and generalize earlier ones.  ...  There are two major challenges in building numerical solvers for nonlinear and degenerate elliptic Partial Differential Equations (PDEs).  ... 
arXiv:1107.5278v3 fatcat:5zp4jnfw25c6hj7bncxd76v5mu

subject index volumes 201 to 210

2007 Journal of Computational and Applied Mathematics  
744 Nonlinear least squares 203 264; 208 331 Nonlinear operator equations 203 279; 206 873 Nonlinear optimization 206 1070 Nonlinear oscillator 207 148 Numerical conformal mapping 209 1 Numerical  ...  coordinates 204 231 Elliptic cylinders 204 231 Elliptic equations 202 414; 204 3; 206 136; 207 301 Elliptic partial differential equations 206 1082 Elliptic problems 202 230; 206 843 Elliptic systems  ... 
doi:10.1016/s0377-0427(07)00502-x fatcat:tovvilkoczetvl423gcjxug7gm

Manifold Learning and Nonlinear Homogenization [article]

Shi Chen, Qin Li, Jianfeng Lu, Stephen J. Wright
2021 arXiv   pre-print
Our framework is applied to a semilinear elliptic equation with oscillatory media and a nonlinear radiative transfer equation; in both cases, significant improvements in efficacy are observed.  ...  We describe an efficient domain decomposition-based framework for nonlinear multiscale PDE problems.  ...  We have described numerical methods that can capture the homogenization limit of nonlinear PDEs with small scales automatically, without analytical prior knowledge.  ... 
arXiv:2011.00568v2 fatcat:544hfcgdonacnlwok2fitbqlke

Page 2754 of Mathematical Reviews Vol. , Issue 2002D [page]

2002 Mathematical Reviews  
In this work, a few classes of related multi-level explicit schemes for solving certain elliptic boundary value problems are considered.  ...  These estimates show that the scheme (1) may be used as an iterative method to solve a stationary equation of the type Ay = f with self-adjoint and positive operator A.  ... 

A boundary value problem for the KdV equation: Comparison of finite-difference and Chebyshev methods

Jan Ole Skogestad, Henrik Kalisch
2009 Mathematics and Computers in Simulation  
Most strategies found in literature for solving nonlinear problems involve a linearization step, usually using Newton's method, which replaces the original nonlinear problem by an iteration process consisting  ...  In this work includes a comparative study of two discretization methods with highly different properties for this equation. v Acknowledgements This thesis was written as a part of my PhD study in applied  ...  Scalability with respect to linear and nonlinear iterations is studied for problems with homogeneous and heterogeneous parameter fields.  ... 
doi:10.1016/j.matcom.2009.06.009 fatcat:6xy56ohmlncojiu4o7qzzppc4q

Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs

Boris N. Khoromskij, Christoph Schwab
2011 SIAM Journal on Scientific Computing  
Such PDEs arise, for example, in the parametric, deterministic reformulation of elliptic PDEs with random field inputs, based for example, on the M -term truncated Karhunen-Loève expansion.  ...  Our approach could be regarded as either a class of compressed approximations of these solution or as a new class of iterative elliptic problem solvers for high dimensional, parametric, elliptic PDEs providing  ...  The principal idea of our approach is the iterative solution of a single coupled system of discrete, multiparametric elliptic equations projected onto the nonlinear manifold of low rank tensor-structured  ... 
doi:10.1137/100785715 fatcat:cflenwndw5h4towm2rizjrqyla

Simultaneous space–time adaptive wavelet solution of nonlinear parabolic differential equations

Jahrul M. Alam, Nicholas K.-R. Kevlahan, Oleg V. Vasilyev
2006 Journal of Computational Physics  
Dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time.  ...  The same time step is used for all spatial locations and all scales: this approach clearly does not fully exploit space-time intermittency.  ...  Partial support for OVV was provided by the National Science Foundation (NSF) under grants No. EAR-0327269 and ACI-0242457 and National Aeronautics and Space Administration (NASA) under grant No.  ... 
doi:10.1016/ fatcat:mazptfrlanay5g54uq6mhpfmum
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