scholarly journals Stochastic Homogenization of Linear Elliptic Equations: Higher-Order Error Estimates in Weak Norms Via Second-Order Correctors

2017 ◽  
Vol 49 (6) ◽  
pp. 4658-4703 ◽  
Author(s):  
Peter Bella ◽  
Benjamin Fehrman ◽  
Julian Fischer ◽  
Felix Otto
Keyword(s):  
Error Estimates ◽  
Elliptic Equations ◽  
Higher Order ◽  
Second Order ◽  
Stochastic Homogenization ◽  
Order Error

scholarly journals First- and second-order error estimates in Monte Carlo integration

2016 ◽  
Vol 208 ◽  
pp. 29-34 ◽  
Author(s):  
R. Bakx ◽  
R.H.P. Kleiss ◽  
F. Versteegen
Keyword(s):  
Monte Carlo ◽  
Error Estimates ◽  
Second Order ◽  
Monte Carlo Integration ◽  
Order Error

scholarly journals First and second order error estimates for the Upwind Source at Interface method

2004 ◽  
Vol 74 (249) ◽  
pp. 103-123 ◽  
Author(s):  
Theodoros Katsaounis ◽  
Chiara Simeoni
Keyword(s):  
Error Estimates ◽  
Second Order ◽  
Order Error

scholarly journals Optimal Homogenization Rates in Stochastic Homogenization of Nonlinear Uniformly Elliptic Equations and Systems

2021 ◽  
Author(s):  
Julian Fischer ◽  
Stefan Neukamm
Keyword(s):  
Error Estimates ◽  
Elliptic Equations ◽  
Regularity Theory ◽  
Optimal Order ◽  
Small Scale ◽  
Elliptic Pdes ◽  
Stochastic Homogenization ◽  
Elliptic Case ◽  
Nonlinear Elliptic ◽  
Spatially Homogeneous

AbstractWe derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on $$\mathbb {R}^d$$ R d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale $$\varepsilon >0$$ ε > 0 , we establish homogenization error estimates of the order $$\varepsilon $$ ε in case $$d\geqq 3$$ d ≧ 3 , and of the order $$\varepsilon |\log \varepsilon |^{1/2}$$ ε | log ε | 1 / 2 in case $$d=2$$ d = 2 . Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence $$\varepsilon ^\delta $$ ε δ . We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order $$(L/\varepsilon )^{-d/2}$$ ( L / ε ) - d / 2 for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) $$C^{1,\alpha }$$ C 1 , α regularity theory is available.

scholarly journals A posteriori error estimates for mixed finite volume solution of elliptic boundary value problems

2017 ◽  
Vol 3 (2) ◽  
pp. 199-217
Author(s):  
Fayssal Benkhaldoun ◽  
Mohammed Seaid ◽  
Amadou Mahamane
Keyword(s):  
Error Estimates ◽  
Finite Volume ◽  
Diffusion Coefficients ◽  
Elliptic Equations ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Second Order ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Second Order Elliptic Equations

AbstractThe major emphasis of this work is the derivation of a posteriori error estimates for the mixed finite volume discretization of second-order elliptic equations. The estimates are established for meshes consisting of simplices on unstructured grids. We consider diffusion problems with nonhomogeneous diffusion coefficients. The error estimates are of residual types and are formulated in the energy semi-norm for a locally postprocessed approximate solutions. The estimates are fully computable and locally efficient that they can serve as indicators for adaptive refinement and for the actual control of the error. Numerical results are shown for two test examples in two space dimensions. It is found that the proposed adaptive mixed finite volume method offers a robust and accurate approach for solving second-order elliptic equations, even when highly nonhomogeneous diffusion coefficients are used in the simulations.

scholarly journals On a class of degenerate elliptic equations

1974 ◽  
Vol 55 ◽  
pp. 181-204 ◽  
Author(s):  
Yoshiaki Hashimoto ◽  
Tadato Matsuzawa
Keyword(s):  
Elliptic Equations ◽  
Higher Order ◽  
Elliptic Operators ◽  
Second Order ◽  
General Boundary ◽  
Degenerate Elliptic Equations ◽  
Boundary Problems ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic

We shall prove in Chapter I the hypoellipticity for a class of degenerate elliptic operators of higher order. Chapter II will be devoted to the consideration of the regularity at the boundary for the solutions of general boundary problems for the equations considered in Chapter I being restricted to the second order.

scholarly journals A Kneser Theorem for Higher order Elliptic Equations

1977 ◽  
Vol 20 (1) ◽  
pp. 1-8 ◽  
Author(s):  
W. Allegretto
Keyword(s):  
Fourth Order ◽  
Elliptic Equations ◽  
Higher Order ◽  
Second Order ◽  
Oscillation Criteria ◽  
Fourth Order Equations ◽  
Oscillation Theorems

The problem of establishing oscillation and non-oscillation criteria for elliptic equations has recently been considered by several authors. Extensive bibliographies may be found in the books by C. A. Swanson, [7], and by K. Kreith, [3].Most of the interest has so far centered on equations of second order with some results also established for fourth order equations. Non-oscillation theorems for higher order equations have recently been established by the author, [1], and by Noussair and Yoshida, [5]. In particular, both in [1] and [5], Kneser-type theorems were established for classes of higher order elliptic equations.

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