13.4. Nonlinear elliptic PDEs

Many nonlinear elliptic PDEs include references to the Laplacian operator.

Example 13.4.1

Recall the micromechanical deflector modeled in a disk by (10.1.2). A fully two-dimensional equivalent is (see [PD06])

(13.4.1)\[\Delta u - \frac{\lambda}{(u+1)^2} = 0.\]

This may be posed on any region, with \(u=0\) specified everywhere on the boundary.

More generally, we want to solve the nonlinear equation

(13.4.2)\[\phi(x,y,u,u_x,u_y,u_{xx},u_{yy}) = 0\]

in the interior of a rectangle \(R\), subject to the Dirichlet condition

(13.4.3)\[u(x,y) = g(x,y)\]

on the boundary of \(R\).

Implementation

In order to solve for as few unknowns as possible, we use a Chebyshev discretization of the domain. The core idea is to formulate collocation equations at the grid points based on discrete approximations of (13.4.2) and (13.4.3). If the PDE is nonlinear, then these equations are also nonlinear and are to be solved by a quasi-Newton iteration. Function 13.4.2 is our implementation.

Function 13.4.2 :  elliptic

Newton’s method to solve an elliptic PDE

 1"""
 2    elliptic(ϕ,g,m,xspan,n,yspan)
 3
 4Solve the elliptic PDE `ϕ`(x,y,u,u_x,u_xx,u_y,u_yy)=0 on the 
 5rectangle `xspan` x `yspan`, subject to `g`(x,y)=0 on the boundary. 
 6Uses `m`+1 points in x by `n`+1 points in y in a Chebyshev 
 7discretization.
 8
 9Returns vectors defining the grid and a matrix of grid solution values.
10"""
11function elliptic(ϕ,g,m,xspan,n,yspan)
12    # Discretize the domain.
13    x,Dx,Dxx = diffcheb(m,xspan)
14    y,Dy,Dyy = diffcheb(n,yspan)
15    X = [ x for x in x, y in y ]
16    Y = [ y for x in x, y in y ]
17    unvec = u -> reshape(u,m+1,n+1)
18    
19    # Identify boundary locations and evaluate the boundary condition.
20    isboundary = trues(m+1,n+1)
21    isboundary[2:m,2:n] .= false
22    idx = vec(isboundary)
23    gb = g.(X[idx],Y[idx])
24
25    # Evaluate the discretized PDE and its Jacobian, with all the
26    # boundary condition modifications applied.
27    function residual(u)
28        U = unvec(u)
29        R = ϕ(X,Y,U,Dx*U,Dxx*U,U*Dy',U*Dyy')
30        @. R[idx] = u[idx] - gb
31        return vec(R)
32    end
33
34    # Solve the equation.
35    u = levenberg(residual,vec(zeros(size(X))))[end]
36    U = unvec(u)
37
38    return function (ξ,η)
39		v = [ chebinterp(x,u,ξ) for u in eachcol(U) ]
40		return chebinterp(y,v,η)
41    end
42
43end

Function 13.4.2 first defines the discretization and then computes all the values of \(g\) at the boundary nodes. It uses Function 4.6.4 as the nonlinear solver, and it translates back and forth between vector and grid shapes for the unknowns. After the discrete PDE is collocated at the grid points, the boundary terms are replaced by the boundary residual.

Lines 38–41, which produce the value returned by Function 13.4.2, provide a function that evaluates the numerical solution anywhere in the domain, as is explained next.

Off-grid evaluation

A Chebyshev grid is clustered close to the boundary of the domain, and the grid values may be accurate even for modest grid sizes. As a result, simple piecewise interpolation to evaluate off the grid, as is done by plotting routines, may be unacceptably inaccurate. Instead, we should use the global polynomial interpolation that is foundational to the Chebyshev spectral method.

Let \(\mathbf{U}\) be a matrix of solution values on the Chebyshev grid, defining a function \(u(x,y)\), and let \((\xi,\eta)\) be a point where we want to evaluate \(u(x,y)\). Column \(\mathbf{u}_j\) of the grid matrix represents values spanning all the \(x_i\) while \(y\) is fixed at \(y_j\). Therefore, we can define an interpolating polynomial \(p_j(x)\) based on the values in \(\mathbf{u}_j\).

Now let \(v_j = p_j(\xi)\) for \(j=1,\ldots,n\). The vector \(\mathbf{v}\) is a discretization of \(u(\xi,y)\) at the Chebyshev nodes in \(y\). It defines an interpolating polynomial \(q(y)\), and finally we have \(u(\xi,\eta)=q(\eta)\). You can think of the total process as reducing one dimension at a time through the action of evaluating a polynomial interpolant at a point.

The function returned by Function 13.4.2 performs interpolation as described above, using a helper function chebinterp (not shown). The helper performs the evaluation of a polynomial interpolant in one variable using a modified implementation of Function 9.2.3 that exploits the barycentric weights for Chebyshev nodes given in (9.3.2).1

Demo 13.4.3

We solve the PDE

\[ \Delta u - \frac{\lambda}{(u+1)^2} = 0 \]

on the rectangle \([0,2.5] \times [0,1]\), with a zero Dirichlet condition on the boundary.

All we need to define are \(\phi\) from (13.4.2) for the PDE, and a trivial zero function for the boundary condition.

λ = 1.5
ϕ = (X,Y,U,Ux,Uxx,Uy,Uyy) -> @. Uxx + Uyy - λ/(U+1)^2;
g = (x,y) -> 0; 

Here is the solution for \(m=15\), \(n=8\).

u = FNC.elliptic(ϕ,g,15,[0,2.5],8,[0,1]);
x = range(0,2.5,length=100)
y = range(0,1,length=50)
U = [u(x,y) for x in x, y in y]
contourf(x,y,U',color=:viridis,aspect_ratio=1,
    xlabel=L"x",ylabel=L"y",zlabel=L"u(x,y)",
    title="Deflection of a MEMS membrane")        
Deflection of a MEMS membrane

In the absence of an exact solution, how can we be confident that the solution is accurate? First, the Levenberg iteration converged without issuing a warning, so we should feel confident that the discrete equations were solved. We can check the boundary values easily. For example,

x = range(0,2.5,length=100)
norm( [u(x,0) - g(x,0) for x in x], Inf )
3.491687516723461e-23

Assuming that we encoded the PDE correctly, the remaining source error is truncation from the discretization. We can estimate that by refining the grid a bit and seeing how much the numerical solution changes.

[ u(x,y) for x in 0.5:0.5:2, y in 0.25:0.25:0.75 ]
4×3 Matrix{Float64}:
 -0.174521  -0.236562  -0.174521
 -0.23255   -0.320498  -0.23255
 -0.23255   -0.320498  -0.23255
 -0.174521  -0.236562  -0.174521
u = FNC.elliptic(ϕ,g,25,[0,2.5],14,[0,1]);
[ u(x,y) for x in 0.5:0.5:2, y in 0.25:0.25:0.75 ]
4×3 Matrix{Float64}:
 -0.174516  -0.236554  -0.174516
 -0.23255   -0.320475  -0.23255
 -0.23255   -0.320475  -0.23255
 -0.174516  -0.236554  -0.174516

The original solution may be accurate to about four digits.

Demo 13.4.4

The steady-state limit of an advection-diffusion equation is

\[ 1 - u_x - 2u_y + \epsilon \, \Delta u = 0. \]

Here we solve it with a homogeneous Dirichlet condition on the square \([-1,1]^2\).

ϕ = (X,Y,U,Ux,Uxx,Uy,Uyy) -> @. 1 - Ux - 2Uy + 0.05*(Uxx + Uyy)
g = (x,y) -> 0
u = FNC.elliptic(ϕ,g,32,[-1,1],32,[-1,1]);
x = y = range(-1,1,length=80)
U = [u(x,y) for x in x, y in y]
contourf(x,y,U',color=:viridis,aspect_ratio=1,
     xlabel=L"x",ylabel=L"y",zlabel=L"u(x,y)",
     title="Steady advection–diffusion")
Steady advection–diffusion
Demo 13.4.5

The stationary Allen–Cahn equation in two dimensions is

\[ u(1-u^2)+\epsilon \, \Delta u = 0. \]

The following defines the PDE and a nontrivial Dirichlet boundary condition for the square \([0,1]^2\).

ϕ = (X,Y,U,Ux,Uxx,Uy,Uyy) -> @. U*(1-U^2) + 0.05*(Uxx + Uyy)  
g = (x,y) -> tanh(5*(x+2y-1));

We solve the PDE and then plot the result.

u = FNC.elliptic(ϕ,g,36,[0,1],36,[0,1]);
x = y = range(0,1,length=80)
U = [u(x,y) for x in x, y in y]
contourf( x,y,U',color=:viridis,aspect_ratio=1,
    xlabel=L"x",ylabel=L"y",zlabel=L"u(x,y)",title="Steady Allen-Cahn" )
Steady Allen-Cahn

Exercises

  1. (a) Solve for the steady state of

    \[ u_t = - u_y - x - 2 + \epsilon ( u_{xx} + u_{yy} ) \]

    for \(\epsilon=1\) in \([-1,1]\times[-1,1]\), subject to a homogeneous Dirichlet boundary condition. Use \(m=n=30\) points and plot the solution.

    (b) Repeat part (a) for \(\epsilon=0.1\), which weakens the influence of diffusion relative to advection.

  2. ⌨ A soap film stretched on a wire frame above the \((x,y)\) plane assumes a shape \(u(x,y)\) of minimum area and is governed by

    \[\begin{align*} \operatorname{div} \, \left( \frac{\operatorname{grad} u}{\sqrt{1 + u_x^2 + u_y^2}} \right) &= 0 \text{ in region $R$},\\ u(x,y) &= g(x,y) \text{ on the boundary of $R$}. \end{align*}\]

    Solve the equation on \([-1,1]^2\) with boundary value \(u(x,y)=\tanh(y-2x)\), and make a surface plot of the result. (Hints: Don’t try to rewrite the PDE. Instead, modify Function 13.4.2 so that ϕ is called with arguments (U,Dx,Dy), and compute the PDE in the form given. Also, since convergence is difficult in this problem, use the boundary data over the whole domain as the initial value for levenberg.)

  3. Modify Function 13.4.2 to solve (13.4.2) on \([a,b] \times [c,d]\) with the mixed boundary conditions

    \[ u = 0, \text{ if } x=a \text{ or } y = d, \qquad \frac{\partial u}{\partial n} = 0, \text{ if } x=b \text{ or } y = c, \]

    where \(\frac{\partial}{\partial n}\) is the derivative in the direction of the outward normal. Either condition can be used at a corner point. (Hint: Define index vectors for each side of the domain.) Apply your solver to the PDE \(\Delta u + \sin(3\pi x) = 0\) on \([0,1]^2\), and make a contour plot of the solution. Why do the level curves intersect two of the sides only at right angles?


1

The interpolation algorithm in Function 13.4.2 is inefficient when \(u\) is to be evaluated on a finer grid, as for plotting. A more careful version could re-use the same values \(v_j = p_j(\xi)\) for multiple values of \(\eta\).