I need to solve an elliptic PDE with pseudo-differential operators, eg. a fractional power of the laplacian. In future work, more general pseudo-differential operators might be considered. Is there any way in order to assemble or approximate the stiffness matrix for such problems in dolfinx?
Hi,
I suggest you take a look at this fairly recent paper:
https://www.sciencedirect.com/science/article/pii/S0898122119303773
The author explains how you can solve an elliptic PDE that contains, e.g., a fractional Laplacian using a rational approximation. What you have to do is: compute the rational approximation (e.g., using the brasil function from the author’s Python package, Welcome to baryrat’s documentation! — baryrat documentation), use the resulting coefficients to define and solve a number of problems involving the shifted operator (no fractional power needed!), and then combine these individual solutions to give you an approximate solution to your original problem.
This is the very short and fast version, it’s explained better in the paper.
Hope that helps (after all this time…)!
Best wishes,
Simon
2 Likes
Hi Simon,
Thanks a lot for your answer. This looks like exactly what I need!
Best,
Bernhard