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Introduction I think there are several questions on this site about ODEs of the form $$(x-a)^2 u''(x) = F(x,u,u')$$ with an initial condition at $x=a$. There is no general guarantee that solutions e …

Interpolation error The overshoots are the unavoidable result of interpolation. NDSolve computes the values of u of the DirichletCondition to machine-precision accuracy and approximates the values o …

Spectral methods I present two general ways to approach a second-order linear BVP of the form $$\gamma(x)\, y''(x) + \beta(x)\, y'(x) + \alpha(x)\, y(x) + \varphi(x) = 0,\ y(0) = y_1,\ y(\infty) = y …

The finite element method can be used on this problem if we make a change of variables to convert the domain $[0, \infty)$ to a finite interval. I believe only MachinePrecision is available in FEM. …

I will address the case of an irregular grid. There are two or three ways to produce a pretty good looking plot, through rescaling/FrameTicks or through interpolation. The ticks method involves resc …

You have to create your own mesh and you have to convert your u and v to mesh interpolations. (In the example in the documentation, NDSolveValue does this itself in constructing uif, vif.) Example: …

Method 1: Construct mesh elements manually We can triangulate a periodic quad-lattice on the surface: n = {180, 20}; (* number of points in each direction *) pts = Table[ g[4. Pi/n[[1]] t, 2. Pi/ …

There is the as yet undocumented FEM function (see this answer, or read the user-interface with Definition after clearing the ReadProtected attribute) ElementMeshPlot3D[valuesOnMeshCoordinates, eleme …

First, there are multiple solutions. Second, the one shown on math.SE has such a narrow range of feasible starting initial conditions fo $\epsilon = 1/10$ that the default shooting method in NDSolve i …

In NDSolve you can only approximate a discontinuity in the spatial domain by interpolation over a discrete set. In FEM, this is the only option and the order of interpolation is restricted to 1 or 2. …

fn = {-Cos[u] (1.2 - Cos[(u - Pi)/2]^6) (0.2 + Cos[10 u]^10), Sin[u] (1.2 - Cos[(u - Pi)/2]^6) (0.2 + Cos[10 u]^10)}; plot = ParametricPlot[fn, {u, 0, 2 Pi}, PlotPoints -> Round[2 Pi (Sqrt@M …

An FEM element-meshing approach. The quality is controlled by the option "MaxCellMeasure" -> {"Length" -> 0.05}. Note that the VertexNormals -> -coords option causes the polygonal sphere to be smoot …

I began this thinking of adding a comment to the answer by bbgodfrey, who points out mathematical fixes to the OP's problem. But each idea led to another test and another idea. Aside from the uncere …

A small remark on the error: CoefficientArrays::poly: -(1 + 3.27432/(1 + 0.092 Ccu)^2) Ccu11498 - (3 Ccu11499)/200 + 1.38465 Ccu$11500 is not a polynomial. >> First note that Ccu is your depende …

The OP's updated example The OP's example exhibits some numerical problems about which the fastidious ToElementMesh and even some System functions complain. Since the OP is dealing with the System` R …