pdenonlin

(Not recommended) Solve nonlinear elliptic PDE problem

pdenonlin is not recommended. Use solvepde instead.

Description

example

u = pdenonlin(model,c,a,f) solves the nonlinear PDE

$-\nabla \cdot \left(c\nabla u\right)+au=f$

with geometry, boundary conditions, and finite element mesh in model, and coefficients c, a, and f. In this context, “nonlinear” means some coefficient in c, a, or f depends on the solution u or its gradient. If the PDE is a system of equations (model.PDESystemSize > 1), then pdenonlin solves the system of equations

$-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$

example

u = pdenonlin(b,p,e,t,c,a,f) solves the PDE with boundary conditions b, and finite element mesh (p,e,t).

example

u = pdenonlin(___,Name,Value), for any previous arguments, modifies the solution process with Name, Value pairs.

[u,res] = pdenonlin(___) also returns the norm of the Newton step residuals res.

Examples

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Solve a minimal surface problem. Because this problem has a nonlinear c coefficient, use pdenonlin to solve it.

Create a model and include circular geometry using the built-in circleg function.

model = createpde;
geometryFromEdges(model,@circleg);

Set the coefficients.

a = 0;
f = 0;
c = '1./sqrt(1+ux.^2+uy.^2)';

Set a Dirichlet boundary condition with value ${x}^{2}$.

boundaryfun = @(region,state)region.x.^2;
applyBoundaryCondition(model,'edge',1:model.Geometry.NumEdges,...
'u',boundaryfun,'Vectorized','on');

Generate a mesh and solve the problem.

generateMesh(model,'GeometricOrder','linear','Hmax',0.1);
u = pdenonlin(model,c,a,f);
pdeplot(model,'XYData',u,'ZData',u) Solve the minimal surface problem using the legacy approach for creating boundary conditions and geometry.

Create the geometry using the built-in circleg function. Plot the geometry to see the edge labels.

g = @circleg;
pdegplot(g,'EdgeLabels','on')
axis equal Create Dirichlet boundary conditions with value ${x}^{2}$.Create the following file and save it on your MATLAB path.

function [qmatrix,gmatrix,hmatrix,rmatrix] = pdex2bound(p,e,u,time)
ne = size(e,2); % number of edges
qmatrix = zeros(1,ne);
gmatrix = qmatrix;
hmatrix = zeros(1,2*ne);
rmatrix = hmatrix;
for k = 1:ne
x1 = p(1,e(1,k)); % x at first point in segment
x2 = p(1,e(2,k)); % x at second point in segment
xm = (x1 + x2)/2; % x at segment midpoint
y1 = p(2,e(1,k)); % y at first point in segment
y2 = p(2,e(2,k)); % y at second point in segment
ym = (y1 + y2)/2; % y at segment midpoint
switch e(5,k)
case {1,2,3,4}
hmatrix(k) = 1;
hmatrix(k+ne) = 1;
rmatrix(k) = x1^2;
rmatrix(k+ne) = x2^2;
end
end

Set the coefficients and boundary conditions.

a = 0;
f = 0;
c = '1./sqrt(1+ux.^2+uy.^2)';
b = @pdex2bound;

Generate a mesh and solve the problem.

[p,e,t] = initmesh(g,'Hmax',0.1);
u = pdenonlin(b,p,e,t,c,a,f);
pdeplot(p,e,t,'XYData',u,'ZData',u) Solve a nonlinear 3-D problem with nontrivial geometry.

Import the geometry from the BracketWithHole.stl file. Plot the geometry and face labels.

model = createpde();
importGeometry(model,'BracketWithHole.stl');
figure
pdegplot(model,'FaceLabels','on','FaceAlpha',0.5)
view(30,30)
title('Bracket with Face Labels') figure
pdegplot(model,'FaceLabels','on','FaceAlpha',0.5)
view(-134,-32)
title('Bracket with Face Labels, Rear View') Set a Dirichlet boundary condition with value 1000 on the back face, which is face 4. Set the large faces 1 and 7, and also the circular face 11, to have Neumann boundary conditions with value g = -10. Do not set boundary conditions on the other faces. Those faces default to Neumann boundary conditions with value g = 0.

applyBoundaryCondition(model,'Face',4,'u',1000);
applyBoundaryCondition(model,'Face',[1,7,11],'g',-10);

Set the c coefficient to 1, f to 0.1, and a to the nonlinear value '0.1 + 0.001*u.^2'.

c = 1;
f = 0.1;
a = '0.1 + 0.001*u.^2';

Generate the mesh and solve the PDE. Start from the initial guess u0 = 1000, which matches the value you set on face 4. Turn on the Report option to observe the convergence during the solution.

generateMesh(model);
u = pdenonlin(model,c,a,f,'U0',1000,'Report','on');
Iteration     Residual     Step size  Jacobian: full
0          7.2059e-01
1          1.3755e-01   1.0000000
2          4.0800e-02   1.0000000
3          1.1344e-02   1.0000000
4          2.2737e-03   1.0000000
5          1.7770e-04   1.0000000
6          1.4199e-06   1.0000000

Plot the solution on the geometry boundary.

pdeplot3D(model,'ColorMapData',u) Input Arguments

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PDE model, specified as a PDEModel object.

Example: model = createpde

PDE coefficient, specified as a scalar, matrix, character vector, character array, string scalar, string vector, or coefficient function. c represents the c coefficient in the scalar PDE

$-\nabla \cdot \left(c\nabla u\right)+au=f$

or in the system of PDEs

$-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$

Example: 'cosh(x+y.^2)'

Data Types: double | char | string | function_handle
Complex Number Support: Yes

PDE coefficient, specified as a scalar, matrix, character vector, character array, string scalar, string vector, or coefficient function. a represents the a coefficient in the scalar PDE

$-\nabla \cdot \left(c\nabla u\right)+au=f$

or in the system of PDEs

$-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$

Example: 2*eye(3)

Data Types: double | char | string | function_handle
Complex Number Support: Yes

PDE coefficient, specified as a scalar, matrix, character vector, character array, string scalar, string vector, or coefficient function. f represents the f coefficient in the scalar PDE

$-\nabla \cdot \left(c\nabla u\right)+au=f$

or in the system of PDEs

$-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$

Example: char('sin(x)';'cos(y)';'tan(z)')

Data Types: double | char | string | function_handle
Complex Number Support: Yes

Boundary conditions, specified as a boundary matrix or boundary file. Pass a boundary file as a function handle or as a file name. A boundary matrix is generally an export from the PDE Modeler app.

Example: b = 'circleb1', b = "circleb1", or b = @circleb1

Data Types: double | char | string | function_handle

Mesh points, specified as a 2-by-Np matrix of points, where Np is the number of points in the mesh. For a description of the (p,e,t) matrices, see Mesh Data as [p,e,t] Triples.

Typically, you use the p, e, and t data exported from the PDE Modeler app, or generated by initmesh or refinemesh.

Example: [p,e,t] = initmesh(gd)

Data Types: double

Mesh edges, specified as a 7-by-Ne matrix of edges, where Ne is the number of edges in the mesh. For a description of the (p,e,t) matrices, see Mesh Data as [p,e,t] Triples.

Typically, you use the p, e, and t data exported from the PDE Modeler app, or generated by initmesh or refinemesh.

Example: [p,e,t] = initmesh(gd)

Data Types: double

Mesh triangles, specified as a 4-by-Nt matrix of triangles, where Nt is the number of triangles in the mesh. For a description of the (p,e,t) matrices, see Mesh Data as [p,e,t] Triples.

Typically, you use the p, e, and t data exported from the PDE Modeler app, or generated by initmesh or refinemesh.

Example: [p,e,t] = initmesh(gd)

Data Types: double

Name-Value Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Jacobian','full'

Approximation of Jacobian, specified as 'full', 'fixed', or 'lumped'.

• 'full' means numerical evaluation of the full Jacobian based on the sparse version of the numjac function. 3-D geometry uses only 'full', any other specification yields an error.

• 'fixed' specifies a fixed-point iteration matrix where the Jacobian is approximated by the stiffness matrix. This is the 2-D geometry default.

• 'lumped' specifies a “lumped” approximation as described in Nonlinear Equations. This approximation is based on the numerical differentiation of the coefficients.

Example: u = pdenonlin(model,c,a,f,'Jacobian','full')

Data Types: char | string

Initial solution guess, specified as a scalar, a vector of characters, or a vector of numbers. A scalar specifies a constant initial condition for either a scalar or PDE system.

For systems of N equations, and a mesh with Np nodes, give a column vector with N*Np components. The nodes are either model.Mesh.Nodes, or the p data from initmesh or meshToPet. See Mesh Data as [p,e,t] Triples.

The first Np elements contain the values of component 1, where the value of element k corresponds to node p(k). The next Np points contain the values of component 2, etc. It can be convenient to first represent the initial conditions u0 as an Np-by-N matrix, where the first column contains entries for component 1, the second column contains entries for component 2, etc. The final representation of the initial conditions is u0(:).

Example: u = pdenonlin(model,c,a,f,'U0','x.^2-y.^2')

Data Types: double | char | string
Complex Number Support: Yes

Residual size at termination, specified as a positive scalar. pdenonlin iterates until the residual size is less than 'Tol'.

Example: u = pdenonlin(model,c,a,f,'Tol',1e-6)

Data Types: double

Maximum number of Gauss-Newton iterations, specified as a positive integer.

Example: u = pdenonlin(model,c,a,f,'MaxIter',12)

Data Types: double

Minimum damping of search direction, specified as a positive scalar.

Example: u = pdenonlin(model,c,a,f,'MinStep',1e-3)

Data Types: double

Print convergence information, specified as 'off' or 'on'.

Example: u = pdenonlin(model,c,a,f,'Report','on')

Data Types: char | string

Residual norm, specified as the p value for Lp norm, or as 'energy'. p can be any positive real value, Inf, or -Inf. The p norm of a vector v is sum(abs(v)^p)^(1/p). See norm.

Example: u = pdenonlin(model,c,a,f,'Norm',2)

Data Types: double | char | string

Output Arguments

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PDE solution, returned as a vector.

• If the PDE is scalar, meaning only one equation, then u is a column vector representing the solution u at each node in the mesh. u(i) is the solution at the ith column of model.Mesh.Nodes or the ith column of p.

• If the PDE is a system of N > 1 equations, then u is a column vector with N*Np elements, where Np is the number of nodes in the mesh. The first Np elements of u represent the solution of equation 1, then next Np elements represent the solution of equation 2, etc.

To obtain the solution at an arbitrary point in the geometry, use pdeInterpolant.

To plot the solution, use pdeplot for 2-D geometry, or see 3-D Solution and Gradient Plots with MATLAB® Functions.

Norm of Newton step residuals, returned as a scalar. For information about the algorithm, see Nonlinear Equations.

Tips

• If the Newton iteration does not converge, pdenonlin displays the error message Too many iterations or Stepsize too small.

• If the initial guess produces matrices containing NaN or Inf elements, pdenonlin displays the error message Unsuitable initial guess U0 (default: U0 = 0).

• If you have very small coefficients, or very small geometric dimensions, pdenonlin can fail to converge, or can converge to an incorrect solution. If so, you can sometimes obtain better results by scaling the coefficients or geometry dimensions to be of order one.

Algorithms

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Nonlinear Equations

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

$u\left(x\right)=\sum {U}_{j}{\varphi }_{j}$

where x represents a 2-D or 3-D point. Then multiply the equation by an arbitrary test function ϕi, integrate on the domain Ω, and use Green's formula and the boundary conditions to obtain

$\begin{array}{l}0=\rho \left(U\right)=\sum _{j}\left(\underset{\Omega }{\int }\left(\left(c\left(x,U\right)\nabla {\varphi }_{j}\left(x\right)\right)\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\nabla {\varphi }_{j}\left(x\right)+a\left(x,U\right){\varphi }_{j}\left(x\right){\varphi }_{i}\left(x\right)\text{\hspace{0.17em}}\right)dx\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{\partial \Omega }{\int }q\left(x,U\right){\varphi }_{j}\left(x\right){\varphi }_{i}\left(x\right)\text{\hspace{0.17em}}ds\right){U}_{j}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{\Omega }{\int }f\left(x,U\right){\varphi }_{i}\left(x\right)\text{\hspace{0.17em}}dx-\underset{\partial \Omega }{\int }g\left(x,U\right){\varphi }_{i}\left(x\right)\text{\hspace{0.17em}}ds\end{array}$

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

$\frac{\partial \rho \left({U}^{\left(n\right)}\right)}{\partial U}\left({U}^{\left(n+1\right)}-{U}^{\left(n\right)}\right)=-\alpha \rho \left({U}^{\left(n\right)}\right)$

where $\alpha$ 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 minU $‖\rho \left(U\right)‖$.

It is well known that for sufficiently small $\alpha$

$‖\rho \left({U}^{\left(n+1\right)}\right)‖<‖\rho \left({U}^{\left(n\right)}\right)‖$

and

${p}_{n}={\left(\frac{\partial \rho \left({U}^{\left(n\right)}\right)}{\partial U}\right)}^{-1}\rho \left({U}^{\left(n\right)}\right)$

is called a descent direction for $‖\rho \left(U\right)‖$, where $‖\cdot ‖$ is the L2-norm. The iteration is

${U}^{\left(n+1\right)}={U}^{\left(n\right)}+\alpha {p}_{n},$

where $\alpha$ ≤ 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:

$‖\rho \left({U}^{\left(n\right)}\right)‖-‖\rho \left({U}^{\left(n\right)}\right)+\alpha {p}_{n}‖\ge \frac{\alpha }{2}‖\rho \left({U}^{\left(n\right)}\right)‖$

which guarantees a reduction of the residual norm by at least 1 – $\alpha$/2. Each step of the line-search algorithm requires an evaluation of the residual $\rho \left({U}^{\left(n\right)}+\alpha {p}_{n}\right)$.

An important point of this strategy is that when U(n) approaches the solution, then $\alpha$→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–1F

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 $\stackrel{˜}{U}$ on a mesh, evaluates the error, and may refine certain triangles. The interpolant of $\stackrel{˜}{U}$ is a very good starting guess for the solution on the refined mesh.

In general the exact Jacobian

${J}_{n}=\frac{\partial \rho \left({U}^{\left(n\right)}\right)}{\partial U}$

is not available. Approximation of Jn by finite differences in the following way is expensive but feasible. The ith column of Jn can be approximated by

$\frac{\rho \left({U}^{\left(n\right)}+\epsilon {\varphi }_{i}\right)-\rho \left({U}^{\left(n\right)}\right)}{\epsilon }$

which implies the assembling of the FEM matrices for the triangles containing grid point i. A very simple approximation to Jn, 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–1F .

This is equivalent to approximating the Jacobian with the stiffness matrix. Indeed, since ρ(U(n)) = KU(n)F, putting Jn = K yields

${U}^{\left(n+1\right)}={U}^{\left(n\right)}-{J}_{n}^{-1}\rho \left({U}^{\left(n\right)}\right)={U}^{\left(n\right)}-{K}^{-1}\left(K{U}^{\left(n\right)}-F\right)={K}^{-1}F$

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 UKU, proceed as follows. The a term has been omitted for clarity, but appears again in the final result.

$\begin{array}{c}\frac{\partial {\left(KU\right)}_{i}}{\partial {U}_{j}}=\underset{\epsilon \to 0}{\mathrm{lim}}\frac{1}{\epsilon }\sum _{l}\left(\underset{\Omega }{\int }c\left(U+\epsilon {\varphi }_{j}\right)\nabla {\varphi }_{l}\nabla {\varphi }_{i}\text{\hspace{0.17em}}dx\left({U}_{l}+\epsilon {\delta }_{l,j}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{\Omega }{\int }c\left(U\right)\nabla {\varphi }_{l}\nabla {\varphi }_{i}\text{\hspace{0.17em}}dx{U}_{l}\right)\\ =\underset{\Omega }{\int }c\left(U\right)\nabla {\varphi }_{j}\nabla {\varphi }_{i}\text{\hspace{0.17em}}dx+\sum _{l}\underset{\Omega }{\int }{\varphi }_{j}\frac{\partial c}{\partial u}\nabla {\varphi }_{l}\nabla {\varphi }_{i}\text{\hspace{0.17em}}dx{U}_{l}\end{array}$

The first integral term is nothing more than Ki,j.

The second term is “lumped,” i.e., replaced by a diagonal matrix that contains the row sums. Since Σjϕj = 1, the second term is approximated by

${\delta }_{i,j}\sum _{l}\underset{\Omega }{\int }\frac{\partial c}{\partial u}\nabla {\varphi }_{l}\nabla {\varphi }_{i}\text{\hspace{0.17em}}dx{U}_{l}$

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 UMU. The derivative of the mapping U→ –F is exactly

$-\underset{\Omega }{\int }\frac{\partial f}{\partial u}{\varphi }_{i}{\varphi }_{j}\text{\hspace{0.17em}}dx$

which is the mass matrix associated with the coefficient ∂f/∂u. Thus the Jacobian of the residual ρ(U) is approximated by

$J={K}^{\left(c\right)}+{M}^{\left(a-{f}^{\prime }\right)}+\text{diag}\left(\left({K}^{\left({c}^{\prime }\right)}+{M}^{\left({a}^{\prime }\right)}\right)U\right)$

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

$\begin{array}{l}-\nabla \cdot \left(c\nabla u\right)+\left(a-f\text{'}\right)u=0\\ -\nabla \cdot \left(c\text{'}\nabla u\right)+a\text{'}u=0\end{array}$

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:

$\stackrel{˜}{K}\stackrel{˜}{U}=\left[\begin{array}{cc}K& {H}^{\prime }\\ H& 0\end{array}\right]\left[\begin{array}{c}U\\ \mu \end{array}\right]=\left[\begin{array}{c}F\\ R\end{array}\right]=\stackrel{˜}{F}$

where the coefficients of $\stackrel{˜}{K}$ and $\stackrel{˜}{F}$ may depend on the solution $\stackrel{˜}{U}$. 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 $\stackrel{˜}{U}$ 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. 