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Iterated numerical homogenization for multi-scale elliptic equations with monotone nonlinearity
[article]
2021
arXiv
pre-print
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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]
2012
Advances in DUNE
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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
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[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
2016
Journal of Scientific Computing
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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]
2020
arXiv
pre-print
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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
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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
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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]
2015
arXiv
pre-print
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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]
2012
arXiv
pre-print
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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
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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]
2021
arXiv
pre-print
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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
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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
2009
Mathematics and Computers in Simulation
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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
2011
SIAM Journal on Scientific Computing
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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
2006
Journal of Computational Physics
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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/j.jcp.2005.10.009
fatcat:mazptfrlanay5g54uq6mhpfmum
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