Solving nonlinear FEM in MATLAB

Question

I try to solve a heat diffusion problem on tetrahedral finite elements with nodal heat sources, which depend on the solution vector, in MATLAB. The nonlinear equation system looks like this:

B U' + A U = q(T)

with B being the heat capactiy matrix, A being the conductivity matrix, q being the source terms and U being the Temperature. I use an Adams-Bashforth/Trapezoid Rule predictor-corrector scheme with a Picard iteration followed by a time step control. The temperature for the source terms is evaluated exactly between the last time step's temperature and the predictor's temperature. Here is a simplified version of the predictor-corrector code. The calculation of the sources is a function.

    % predictor
    K0 = t(n)-t(n-1);              
    Upre(dirichlet) = u_d_t(coordinates(dirichlet,:));
    Upre(FreeNodes) = U(FreeNodes,n) + (dt/2)*((2+dt/K0)*U_dt(FreeNodes,3) - (dt/K0)*U_dt(FreeNodes,1));      % predictor step
    Uguess = Upre;     % save as initial guess for Picard iteration

    % corrector with picard iteration
    while res >= picard_tolerance 
        T_theta = Uguess*theta + (1-theta)*U(:,n);
        b = q(T_theta);
        % Building right-hand side vector (without Dirichlet boundary conditions yet)
        rhs = ((2/dt)*B*U(:,n) + B*U_dt(:,1))+b;

        % Applying Dirichlet Boundary Conditions to the Solution Vector
        Ucor(dirichlet) = u_d_t(coordinates(dirichlet,:));
        rhs = rhs - ((2/dt)*B+A)*Ucor;

        % Solving the linearized system using the backslash operator
        % P*U(n+1) = f(Un) => U(n+1) = P\f(Un)
        Ucor(FreeNodes) = ((2/dt)*B(FreeNodes,FreeNodes)+A(FreeNodes,FreeNodes))\rhs(FreeNodes);
        res = norm(Uguess-Ucor);
        Uguess = Ucor;
        U(:,n+1) = Ucor;
    end

As you can see I use the backslash operator to solve the system. The non-linearity of the system should not be to bad. However, with increasing time steps the picard method converges more slowly and eventually stops to converge altogether. I need much bigger time steps though, so I put the whole corrector step into a function and tried to solve it with fsolve instead to see if I achieve quicker convergence. Unfortunately fsolve seems to never even finish the first time step. I suppose I did not configure the options for fsolve correctly. Can anyone tell me, how to configure fsolve for large sparse nonlinear systems ( We are talking about thousands upt to hundredthousands of equations). Or is there maybe a better solution than fsolve for this problem? Help and - as I am not an expert or computational engineer - explicit advice is greatly appreciated!

Answer 1

In my experience, non-linear equations are solved by linearizing to solve for a temperature increment and iterating to convergence using something like Newton Raphson solver. So if you're using an implicit integration schema, you have an outer time step solution with an inner non-linear solution for the temperature step over the time step.