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The explanation is in the error message (emphasis mine): ToElementMesh::fememins: The mesh elements are not valid. A set of valid mesh element incidents needs to be positive integers and be able t …

The OP's simpler mesh has the form mesh (* ElementMesh[{{-0.707107, 0.707107}, {-1., -0.11547}, {-0.707107, 0.707107}}, {HexahedronElement["<" 64 ">"], HexahedronElement["<" 64 ">"], Hexah …

Sorry just to drop code, but I'm bit too busy at the moment to write up a presentation. Hopefully the code with still be helpful. Ask questions and I'll try to address them later. The testing setup …

You could use a simpler test function for the sake of illustration. Let's take the following function over the implied rectangle, which can integrated exactly for the sake of comparison with the appro …

No, interpolating vectors is possible, but I think that when the data points form an unstructured input grid, then interpolation is restricted to scalar. At least I didn't succeed. However on a recta …

You can build one out of region functions, at least for mesh regions. One could always discretize other regions, but the answer will be approximate and probably not highly accurate. More efficient …

Here's a pretty neat way, based on the new FEM utilities for dealing with meshes, that works automatically for dimension 2 and 3, positively oriented, convex, simplex-element meshes. (More precisely, …

Maybe this can get you started. It assumes the point {x, y} is on the boudary and that the region is described by inequalities involving either <= or >= that are returned by RegionMember. normal[reg …

Somewhat manual construction: << NDSolve`FEM` mesh = ToElementMesh[Disk[], MaxCellMeasure -> Infinity]; tl = Range[0, 1, 0.2]; mesh3 = MeshOrderAlteration[ With[{gc = ElementMeshToGraphicsComplex@ …

Let b1, b2 map the unit circle to the inner, outer boundaries respectively. You might need to handle a list of points with Map (e.g., b1 /@ bdy) depending on b1 and b2. Then here is a simple way: N …

Here's one way. Note NDSolve would have returned an ElementMeshInterpolation, so some of these steps should be unnecessary in your use-case. Needs["NDSolve`FEM`"]; cp = CountryData["China", "Polygo …

Perhaps this is what you want: The only trick is scaling the coordinates of each point to lie on the sphere inscribed in the cube symmetric about the origin circumscribing the point. cube = ToElemen …

It can sometimes help out if you Reduce your implicit description of the region first. ToElementMesh[ ImplicitRegion[! (x^2 + y^2 > RLiner^2 && x^2 + y^2 < RRib^2 && 0 < y < x*Tan[Theta2] …

As shown in the docs for NDEigensystem, the proper way to specify a homogeneous boundary condition with NeumannValue is to add it to the linear operator (only homogeneous boundary conditions are suppo …

If the FEM method suggested by @user21 does not work, here is one way to interpolate between the line solutions. Since you've got lines in terms of the global variable t, we have to take some care wi …