suppose that I want to solve L(u)=f, then we divide the domain into elements, for each element K, we solve L(b_{i})=-L(N_{i}) and L(b_{f})=f, with boundary 0 on K,where N_{i} is the basis functions for coarse scale space,then we use b_{i} to represent basis of fine scale and we add this to N_{i} to form a new basis without analytical expression to form stiff matrix, can FEnicsx natively support this?
I’ve done all my VMS work with FEniCS so the answer is certainly yes. In fact, I’ve found that FEniCS is the perfect tool for this, as it operates at the level of variational formulations, which are key in VMS.
Towards the end of my PhD I wrote this code for computing fine-scale Green’s functions and fine-scale closure function: Stein Stoter / fine scale closure functions · GitLab , used for the computations in this article: Redirecting (I’m afraid I cannot offer any support for the code in this repository though, it’s quite out dated compared to FEniCSx).
I’ll admit that your notation/interpretation is not entirely how I view VMS though. It seems like you have a very speficic idea in mind for reducing the fine-scale equations to element local statements. That’s not entirely formally VMS.
Using mathematical notation by putting your equations inbetween dollar signs would help conveying your ideas: L(u)=f
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