scholarly journals A New Estimate for the Homogenization Method for Second-Order Elliptic Problem with Rapidly Oscillating Periodic Coefficients

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Xiong Liu ◽  
Wenming He
Keyword(s):  
Elliptic Problem ◽  
Homogenization Method ◽  
Second Order ◽  
Homogenization Theory ◽  
High Demand ◽  
Periodic Coefficients ◽  
Order Elliptic Problem ◽  
Multiscale Homogenization

In this paper, we will investigate a multiscale homogenization theory for a second-order elliptic problem with rapidly oscillating periodic coefficients of the form ∂ / ∂ x i a i j x / ε , x ∂ u ε x / ∂ x j = f x . Noticing the fact that the classic homogenization theory presented by Oleinik has a high demand for the smoothness of the homogenization solution u 0 , we present a new estimate for the homogenization method under the weaker smoothness that homogenization solution u 0 satisfies than the classical homogenization theory needs.

On the error estimates of a hybridizable discontinuous Galerkin method for second-order elliptic problem with discontinuous coefficients

2019 ◽  
Vol 40 (2) ◽  
pp. 1577-1600
Author(s):  
Gang Chen ◽  
Jintao Cui
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Elliptic Problem ◽  
Posteriori Error ◽  
Discontinuous Coefficients ◽  
Second Order ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Hybridizable Discontinuous Galerkin ◽  
Order Elliptic Problem

Abstract Hybridizable discontinuous Galerkin (HDG) methods retain the main advantages of standard discontinuous Galerkin (DG) methods, including their flexibility in meshing, ease of design and implementation, ease of use within an $hp$-adaptive strategy and preservation of local conservation of physical quantities. Moreover, HDG methods can significantly reduce the number of degrees of freedom, resulting in a substantial reduction of computational cost. In this paper, we study an HDG method for the second-order elliptic problem with discontinuous coefficients. The numerical scheme is proposed on general polygonal and polyhedral meshes with specially designed stabilization parameters. Robust a priori and a posteriori error estimates are derived without a full elliptic regularity assumption. The proposed a posteriori error estimators are proved to be efficient and reliable without a quasi-monotonicity assumption on the diffusion coefficient.

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