How to solve local problem for each element cell of the mesh?

[dokken]

You could use something along the lines of:

Cell-wise assembly in v10.0.0

This is due to a slight change in the API, where a void ptr is passed as the last argument, to make it possible to pass arbitrary data to (custom) kernels. (Ref: Add void* to tabulate_tensor kernel by sclaus2 · Pull Request #753 · FEniCS/ffcx · GitHub) By changing to self.kernel(ffi_fb(A_local), ffi_fb(coeffs), ffi_fb(self.consts), ffi_fb(geometry), ffi_fb(facet_index), ffi_fb(facet_perm), self.ffi.NULL,) the code runs.

where you make a local assembler for lhs and rhs and solve for that (ref: Invert DG block diagonal matrix - #5 by rosilho)