A localized orthogonal decomposition strategy for hybrid discontinuous Galerkin methods

Abstract

We formulate and analyze a multiscale method for an elliptic problem with an oscillatory coefficient based on a skeletal (hybrid) formulation. More precisely, we employ hybrid discontinuous Galerkin approaches and combine them with the localized orthogonal decomposition methodology to obtain a coarse-scale skeletal method that effectively includes fine-scale information. This work is the first step in reliably merging hybrid skeletal formulations and localized orthogonal decomposition to unite the advantages of both strategies. Numerical experiments are presented to illustrate the theoretical findings.

Figures (2)

• Figure 1: Numerical experiments: Checkerboard case in the left column, random case in the middle column, and channel case in the right column. The structure of the diffusion coefficient is illustrated in the top row, the values in black and white cells are given in the second row, and the error plots are presented in the third row. Here, the red dashed lines represent $\ell = 2$, while the blue solid lines correspond to $\ell = 3$, and the black lines refer to $\ell = 4$. Circular marks refer to the $L^2$ error, and crosses refer to the energy error. The green graphs illustrate simulations with $\ell = 5$, where the contrast has been multiplied by a factor of 10: In the first example, the value of white pixels is replaced by 100, in the second one by 10, and in the third one by 1000. Here, squares refer to the $L^2$ error, and diamonds to the energy error in this case.
• Figure 2: Illustration of a non-convex Lipschitz domain (left) and a non-convex square with crack (right) with possible decomposition.

Theorems & Definitions (32)

• Lemma 3.1
• proof
• Remark 3.2: Verification for the RT-H methods
• Remark 3.3: Other methods
• Lemma 3.4
• proof
• Lemma 3.5
• proof : Proof of Lemma \ref{['LEM:uih_error']}
• Theorem 5.1: Error of the prototypical method
• Remark 5.2