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The low-level Partial Differential Equation Toolbox™ functions are aimed at solving linear equations. Since many interesting computational problems are nonlinear, the software contains a nonlinear solver built on top of the assempde function.
Note Before solving a nonlinear elliptic PDE, from the Solve menu in the PDE app, select Parameters. Then, select the Use nonlinear solver check box and click OK. At the command line, use pdenonlin instead of assempde. The parabolic and hyperbolic functions automatically detect when they need to use a nonlinear solver. |
The basic idea is to use Gauss-Newton iterations to solve the nonlinear equations. Say you are trying to solve the equation
r(u) = –∇ · (c(u)∇u) + a(u)u - f(u) = 0.
In the FEM setting you solve the weak form of r(u) = 0. Set as usual
$$u(x)={\displaystyle \sum {U}_{j}}{\varphi}_{j}$$
then, multiply the equation by an arbitrary test function ϕ_{i}, integrate on the domain Ω, and use Green's formula and the boundary conditions to obtain
$$\begin{array}{l}0=\rho \left(U\right)={\displaystyle \sum _{j}({\displaystyle \underset{\Omega}{\int}\left(c\left(x,U\right)\nabla {\varphi}_{j}(x)\right)\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\nabla {\varphi}_{j}(x)+a\left(x,U\right){\varphi}_{j}(x){\varphi}_{i}(x)\text{\hspace{0.17em}}dx}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\displaystyle \underset{\partial \Omega}{\int}q\left(x,U\right){\varphi}_{j}(x){\varphi}_{i}(x)\text{\hspace{0.17em}}ds}){U}_{j}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\displaystyle \underset{\Omega}{\int}f\left(x,U\right){\varphi}_{i}(x)\text{\hspace{0.17em}}dx}-{\displaystyle \underset{\partial \Omega}{\int}g\left(x,U\right){\varphi}_{i}(x)\text{\hspace{0.17em}}ds}\end{array}$$
which has to hold for all indices i.
The residual vector ρ(U) can be easily computed as
ρ(U) = (K + M + Q)U – (F + G)
where the matrices K, M, Q and the vectors F and G are produced by assembling the problem
–∇ · (c(U)∇u) + a(U)u = f(U).
Assume that you have a guess U^{(n)} of the solution. If U^{(n)} is close enough to the exact solution, an improved approximation U^{(n+1)} is obtained by solving the linearized problem
$$\frac{\partial \rho \left({U}^{(n)}\right)}{\partial U}\left({U}^{(n+1)}-{U}^{(n)}\right)=-\alpha \rho \left({U}^{(n)}\right),$$
where α is a positive number. (It is not necessary that ρ(U) = 0 have a solution even if ρ(u) = 0 has.) In this case, the Gauss-Newton iteration tends to be the minimizer of the residual, i.e., the solution of min_{U} $$\Vert \rho (U)\Vert $$.
It is well known that for sufficiently small α
$$\Vert \rho \left({U}^{(n+1)}\right)\Vert <\Vert \rho \left({U}^{(n)}\right)\Vert $$
and
$${p}_{n}={\left(\frac{\partial \rho \left({U}^{(n)}\right)}{\partial U}\right)}^{-1}\rho \left({U}^{(n)}\right)$$
is called a descent direction for $$\Vert \rho (U)\Vert $$, where $$\Vert \cdot \Vert $$ is the L_{2}-norm. The iteration is
U^{(n+1)} = U^{(n)} + αp_{n },
where α ≤ 1 is chosen as large as possible such that the step has a reasonable descent.
The Gauss-Newton method is local, and convergence is assured only when U^{(0)} is close enough to the solution. In general, the first guess may be outside the region of convergence. To improve convergence from bad initial guesses, a damping strategy is implemented for choosing α, the Armijo-Goldstein line search. It chooses the largest damping coefficient α out of the sequence 1, 1/2, 1/4, . . . such that the following inequality holds:
$$\Vert \rho \left({U}^{(n)}\right)\Vert -\Vert \rho \left({U}^{(n)}\right)+\alpha {p}_{n}\Vert \ge \frac{\alpha}{2}\Vert \rho \left({U}^{(n)}\right)\Vert $$
which guarantees a reduction of the residual norm by at least 1 – α/2. Each step of the line-search algorithm requires an evaluation of the residual ρ(U^{(n)} + αp_{n}).
An important point of this strategy is that when U^{(n)} approaches the solution, then α→1 and thus the convergence rate increases. If there is a solution to ρ(U) = 0, the scheme ultimately recovers the quadratic convergence rate of the standard Newton iteration.
Closely related to the preceding problem is the choice of the initial guess U^{(0)}. By default, the solver sets U^{(0)} and then assembles the FEM matrices K and F and computes
U^{(1)} = K^{–1}F
The damped Gauss-Newton iteration is then started with U^{(1)}, which should be a better guess than U^{(0)}. If the boundary conditions do not depend on the solution u, then U^{(1)} satisfies them even if U^{(0)} does not. Furthermore, if the equation is linear, then U^{(1)} is the exact FEM solution and the solver does not enter the Gauss-Newton loop.
There are situations where U^{(0)} = 0 makes no sense or convergence is impossible.
In some situations you may already have a good approximation and the nonlinear solver can be started with it, avoiding the slow convergence regime. This idea is used in the adaptive mesh generator. It computes a solution $$\tilde{U}$$ on a mesh, evaluates the error, and may refine certain triangles. The interpolant of $$\tilde{U}$$ is a very good starting guess for the solution on the refined mesh.
In general the exact Jacobian
$${J}_{n}=\frac{\partial \rho \left({U}^{(n)}\right)}{\partial U}$$
is not available. Approximation of J_{n} by finite differences in the following way is expensive but feasible. The ith column of J_{n} can be approximated by
$$\frac{\rho \left({U}^{(n)}+\epsilon {\varphi}_{i}\right)-\rho \left({U}^{(n)}\right)}{\epsilon}$$
which implies the assembling of the FEM matrices for the triangles containing grid point i. A very simple approximation to J_{n}, which gives a fixed point iteration, is also possible as follows. Essentially, for a given U^{(n)}, compute the FEM matrices K and F and set
U^{(n+1)} = K^{–1}F .
This is equivalent to approximating the Jacobian with the stiffness matrix. Indeed, since ρ(U^{(n)}) = KU^{(n)} – F, putting J_{n} = K yields
$${U}^{(n+1)}={U}^{(n)}-{J}_{n}^{-1}\rho \left({U}^{(n)}\right)={U}^{(n)}-{K}^{-1}\left(K{U}^{(n)}-F\right)={K}^{-1}F.$$
In many cases the convergence rate is slow, but the cost of each iteration is cheap.
The Partial Differential Equation Toolbox nonlinear solver also provides for a compromise between the two extremes. To compute the derivative of the mapping U→KU, proceed as follows. The a term has been omitted for clarity, but appears again in the final result.
$$\begin{array}{c}\frac{\partial {\left(KU\right)}_{i}}{\partial {U}_{j}}=\underset{\epsilon \to 0}{\mathrm{lim}}\frac{1}{\epsilon}{\displaystyle \sum _{l}({\displaystyle \underset{\Omega}{\int}c\left(U+\epsilon {\varphi}_{j}\right)\nabla {\varphi}_{l}\nabla {\varphi}_{i}\text{\hspace{0.17em}}dx\left({U}_{l}+\epsilon {\delta}_{l,j}\right)}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\displaystyle \underset{\Omega}{\int}c\left(U\right)\nabla {\varphi}_{l}\nabla {\varphi}_{i}\text{\hspace{0.17em}}dx{U}_{l}})\\ ={\displaystyle \underset{\Omega}{\int}c\left(U\right)\nabla {\varphi}_{j}\nabla {\varphi}_{i}\text{\hspace{0.17em}}dx}+{\displaystyle \sum _{l}{\displaystyle \underset{\Omega}{\int}{\varphi}_{j}\frac{\partial c}{\partial u}\nabla {\varphi}_{l}\nabla {\varphi}_{i}\text{\hspace{0.17em}}dx{U}_{l}}}.\end{array}$$
The first integral term is nothing more than K_{i,j}.
The second term is "lumped," i.e., replaced by a diagonal matrix that contains the row sums. Since Σ_{j}ϕ_{j} = 1, the second term is approximated by
$${\delta}_{i,j}{\displaystyle \sum _{l}{\displaystyle \underset{\Omega}{\int}\frac{\partial c}{\partial u}\nabla {\varphi}_{l}\nabla {\varphi}_{i}\text{\hspace{0.17em}}dx{U}_{l}}}$$
which is the ith component of K^{(c')}U, where K^{(c')} is the stiffness matrix associated with the coefficient ∂c/∂u rather than c. The same reasoning can be applied to the derivative of the mapping U→MU. The derivative of the mapping U→ –F is exactly
$$-{\displaystyle \underset{\Omega}{\int}\frac{\partial f}{\partial u}{\varphi}_{i}{\varphi}_{j}\text{\hspace{0.17em}}dx}$$
which is the mass matrix associated with the coefficient ∂f/∂u. Thus the Jacobian of the residual ρ(U) is approximated by
$$J={K}^{(c)}+{M}^{(a-{f}^{\prime})}+\text{diag}\left(\left({K}^{({c}^{\prime})}+{M}^{({a}^{\prime})}\right)U\right)$$
where the differentiation is with respect to u, K and M designate stiffness and mass matrices, and their indices designate the coefficients with respect to which they are assembled. At each Gauss-Newton iteration, the nonlinear solver assembles the matrices corresponding to the equations
$$\begin{array}{l}-\nabla \cdot (c\nabla u)+(a-f\text{'})u=0\\ -\nabla \cdot (c\text{'}\nabla u)+a\text{'}u=0\end{array}$$
and then produces the approximate Jacobian. The differentiations of the coefficients are done numerically.
In the general setting of elliptic systems, the boundary conditions are appended to the stiffness matrix to form the full linear system:
$$\tilde{K}\tilde{U}=\left[\begin{array}{cc}K& {H}^{\prime}\\ H& 0\end{array}\right]\left[\begin{array}{c}U\\ \mu \end{array}\right]=\left[\begin{array}{c}F\\ R\end{array}\right]=\tilde{F},$$
where the coefficients of $$\tilde{K}$$ and $$\tilde{F}$$ may depend on the solution $$\tilde{U}$$. The "lumped" approach approximates the derivative mapping of the residual by
$$\left[\begin{array}{cc}J& {H}^{\prime}\\ H& 0\end{array}\right]$$
The nonlinearities of the boundary conditions and the dependencies of the coefficients on the derivatives of $$\tilde{U}$$ are not properly linearized by this scheme. When such nonlinearities are strong, the scheme reduces to the fix-point iteration and may converge slowly or not at all. When the boundary conditions are linear, they do not affect the convergence properties of the iteration schemes. In the Neumann case they are invisible (H is an empty matrix) and in the Dirichlet case they merely state that the residual is zero on the corresponding boundary points.