I am trying to solve a set of nonlinear time-independent PDEs, e.g., $$L{[\bf{u}]=0}……(1)$$ where $L$ is a nonlinear differential operator and $\bf{u}$ is the unknowns. The specific form of $L$ is too complicated to be given here. It is hard to numerically solve eq.(1) directly.
A so-called 'dynamic relaxation (RD)' method is employed in the literature to work around, see e.g., paper 1 and paper 2. The basic idea is to convert the time-independent eq.(1) into a time-dependent one, like $${\bf{M}}\frac{\partial^2{\bf{u}}}{\partial{t}^2}+{\bf{C}}\frac{\partial{\bf{u}}}{\partial{t}}=L{[\bf{u}]}……(2) $$ where the ${\bf{M}}$ and ${\bf{C}}$ are pseudo- mass and damp matrixes and don't need to have realistic physical meanings. As time elapses, the solution of eq.(2) will ultimately evolve into the solution of eq.(1). The problem is how to assign values to ${\bf{M}}$ and ${\bf{C}}$, so that this evolving process can be as fast as possible and the computational cost can be minimized.
In the previous mentioned paper1 , eq.(1) is firstly discreted by fininte element method, resulting in a set of algebraic equations, like $${\bf{Ku=0}}……(3) $$ Then, the pseudo- mass and damp terms are added according to the coefficient matrix of these algebraic equations, i.e., $${\rho\bf{D}}\frac{\partial^2{\bf{u}}}{\partial{t}^2}+{c\bf{D}}\frac{\partial{\bf{u}}}{\partial{t}}+{\bf{Ku}=0}……(4) $$ where, $\bf{D}$ is the main diagonal terms of $\bf{K}$ and $\rho$ and $c$ are the proportion constants.
I was wondering if it is possible to determine a set of proper values of ${\bf{M}}$ and ${\bf{C}}$ only through eq.(1) before finite element discrete, because I want to employ the weak-form-pde module of comsol to perform the computation, where only the weak form is needed. Actually, I have tried to let ${\bf{M}=0}$ and ${\bf{C}=I}$. It works well for some cases, but for others the evolement is very slow.
This post may not seem to be so mathematically. Please tell me if there is anything unclear.