How to refine FEM mesh for steady state 1D problem?

Question

I am solving a system of 3 steady state 1D ODEs, however the system consists of component that has sharp changes at x = 1, which is very hard to capture. Is there a way to specifically increase the mesh density around the region where sharp changes occur? Here is the line that I call to apply ParametricNDSolveValue.

solutions = ParametricNDSolveValue[{ODEs, BCs}, {a, b, phi}, {x, 0, 4}, {miumaxA, miumaxB}] ;

One way I think of is to define a mesh by making mesh points concentrated around x=1. Here is the code I used to create a list x-coordinates (named xlist) that represent the mesh points that are concentrated around x=1.

(*To create a list of x-coordinates for 1D mesh*)
k = 50;
m = 100;
f[x_] = k*(0.5*(1 - Tanh[m*(x - 1)]) + 0.5*m*Sech[0.1*m*(x - 1)]^2 + 
     0.4);
d[x_] = 1/f[x];
Plot[f[x], {x, 0, 4}, PlotRange -> {0, Full}]
xlist = {0};
xend = xlist[[Length[xlist]]];
coord = {{1}};
i = 1;
While[xend < 4,
 i = i + 1;
 coord = Join[coord, {{i}}];
 xend = xend + d[xend];
 xlist = Insert[xlist, xend, (Length[xlist] + 1)];
 ]
ReplacePart[xlist, Length[xlist] -> 4];

However, I am not sure of how to convert the list of x coordinates to a mesh and apply into ParametricNDSolveValue, as I am fairly new into mathematica modelling. Alternatively, is there other way to refine the mesh?

Answer 1

If you know the discretization xlist of variable x you might create a mesh with

Needs["NDSolve`FEM`"]
xmesh = ToElementMesh[Map[{#} &, xlist]] 

Unfortunatesly you didn't provide your ODEs and BCs, but next step would be

solutions = ParametricNDSolveValue[{ODEs, BCs}, {a, b, phi},Element[x, xmesh], {miumaxA, miumaxB}]

Hope it helps!

Answer 2

You can use ToGradedMesh for this.

Needs["NDSolve`FEM`"]
mesh = ToGradedMesh[{
    {Line[{{0}, {1}}], <|"Alignment" -> "Right"|>},
    {Line[{{1}, {4}}], <|"Alignment" -> "Left"|>}
    }];
MeshRegion[mesh]

With the various options you can refine the mesh further.