MultiscaleFEM.jl

Summary

• Contains the implementation of the High-order multiscale methods. The folder HigherOrderMS contains the 1D-implementation in one-dimension, and HigherOrderMS_2d, the 2D implementation.

Code

1D:

The two examples are located in the main folder HigherOrderMS/. Run global-corrected-ms-method.jl for the method with the additional corrections on the fine scale, i.e., $D\tilde{z}_H \in W_h \subset H^1_0(\Omega) \cap \ker \Pi_H$. This would be MS Method + Global Method (Corrections). Run local-corrected-ms-method.jl for the method with the basis for the additional corrrection. This would be MS Method + MS Method (Corrections) for the additional corrections. Here is a summary:

File	Additional Corrections	Stabilization
global-corrected-ms-method.jl	Yes (Global)	No
local-corrected-ms-method.jl	Yes (Local)	No

Results of the Stabilized Multiscale Method for the Poisson equation:

Oscillatory Diffusion (p=1)	Oscillatory Diffusion (p=3)

2D:

The examples are located in HigherOrderMS_2d/examples/2d_examples/. Run 2d_Heat.jl for the method without the additional corrections. So, this would just be the MS Method $\pm$ Stabilization. Run 2d_Heat_Corrected.jl for the method with the additional corrections with/without stabilization. This would be method that we are interested in: MS Method + MS Method (Corrections) $\pm$ Stabilization. I left comments about how to enable/disable the stabilization part. Here is a summary:

File	Additional Corrections	Stabilization
2d_Heat.jl	No	Yes
2d_Heat_Corrected.jl	Yes	Yes

Examples pictures the stabilized multiscale methods for the Poisson Problem and the Heat Equation:

Poisson Equation	Heat Equation

What's missing?

• In 1D, the stabilization part is available only in the Poisson problem code 1d_rate_of_convergence_Poisson.jl.
• High-precision arithmetic for 2D.
• Better Documentation.
• Merge the 1D and 2D implementation into a single package. The 2D version is already in the form of a package, and the 1D version simply needs to be moved in. (Optional for now.)

References

• Målqvist, A. and Peterseim, D., 2020. Numerical homogenization by localized orthogonal decomposition. Society for Industrial and Applied Mathematics.
• Maier, R., 2021. A high-order approach to elliptic multiscale problems with general unstructured coefficients. SIAM Journal on Numerical Analysis, 59(2), pp.1067-1089.
• Abdulle, A. and Henning, P., 2017. Localized orthogonal decomposition method for the wave equation with a continuum of scales. Mathematics of Computation, 86(304), pp.549-587.
• Dong, Z., Hauck, M., & Maier, R. (2023). An improved high-order method for elliptic multiscale problems. SIAM Journal on Numerical Analysis, 61(4), 1918-1937.
• Krumbiegel, F., & Maier, R. (2024). A higher order multiscale method for the wave equation. IMA Journal of Numerical Analysis, drae059.