I am solving a parabolic equation in the form:
$$ \left( {M \over\tau_j} + A \right) u^{j+1} = M f^j + {u^j \over \tau_j}, $$
where $A$ and $f$ are a dense stiffness matrix and the right hand side of the fractional diffusion problem from this paper:
[Acosta, Gabriel & Bersetche, Francisco & Borthagaray, Juan. (2017). A short FE implementation for a 2d homogeneous Dirichlet problem of a Fractional Laplacian. Computers & Mathematics with Applications. 784-816. 10.1016/j.camwa.2017.05.026. ]
$M$ is the matrix of mass, $u^j$ is the solution at a previous time step and $u^{j+1}$ is the vector of unknowns. $\tau_j$ is a number.
I have previously solved the non-parabolic problem $Au = B$ with LU decomposition through lapack's dgesv
function.
$M$ is a sparse matrix with around 5 elements per row. It could also be reduced to a diagonal matrix (lumping of mass) by summing all coefficients to the diagonal element.
So my question is: is it possible to calculate $A = LU$ once and then only update the $LU$ factorization with the $M \over \sigma_j$ for each step?
I have a vague memory that I found a lecture notes or a textbook online about it a few months ago. Alas my google-fu is failing me right now. I tried googling for "parabolic equations solution with LU" and a few other variants about parabolic equations.