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So far what I understand is that two neighbouring elements are conformal if their edges and faces match exactly, whereas with non-conformal elements this is not the case. For instance, h-refinement gives rise to hanging nodes, and h-nonconforming elements.

But then there is the concept of p-nonconforming. Note sure, but I think this applies for instance in DGFEM, where the weight functions are the roots of a polynomial and which do not match at the boundaries?

I would appreciate it if someone offered a simple explanation for this, particularly in the context of typical structured grids used in DGFEM, FEM and FVM.

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The concept of $p$-nonconforming meshes can also apply to continuous FEM, not just DG. For continuous FEM, continuity is enforced by enforcing a single set of degrees of freedom on a face shared between two elements. If those elements have varying polynomial degrees, the trace space on the face must be made the same. This may be done by restricting the higher degree element to a lower degree trace (referred to as the minimum rule) or by increasing the degree of the trace space for the lower degree element (referred to as the maximum rule).

For DG, $p$-nonconforming meshes are much simpler to implement, as elements are coupled together weakly through a unique interface flux. The method behaves is pretty much identically so long as the computation of integrals of this flux are computed with sufficiently accurate quadrature (though implementation details may change between uniform $p$ and variable $p$ codes).

I'm less familiar with FVM, but I think $p$-nonconformity may depend on what type of high $p$ FVM you use (wider stencil, compact/multi-resolution, etc).

Edit: As noted by @Christian Waluga, nonconformity here refers to meshes, not FEM.

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  • $\begingroup$ Thanks Jesse. Am I right in saying that for DG, when one refers to non-conformal elements, they are referring to p-nonconformity due to the discontinuity at the element faces as a result of mismatching shape functions at the faces (for instance if the shape functions are located at the roots of a Legendre polynomial). $\endgroup$
    – melody
    Commented Dec 1, 2015 at 15:31
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    $\begingroup$ It depends - they can be referring to $h$-nonconformity for DG as well. In both $h$ and $p$ nonconformity, the basis functions at faces don't match and some work has to be done to pair them. $\endgroup$
    – Jesse Chan
    Commented Dec 1, 2015 at 17:58
  • $\begingroup$ How can a continuous FEM be nonconforming as you state in your answer? I always thought 'conforming' means that the discrete space is a subset of the space in which the continuous problem is posed. So for H1 problems any continuous FEM should be considered 'conforming', regardless of the (possibly varying) polynomial degree of the ansatz functions. Same applies to nonmatching h-refinements if hanging nodes are treated by additional constraints. $\endgroup$
    – Christian Waluga
    Commented Dec 1, 2015 at 18:17
  • $\begingroup$ I think it's a difference of terminology - nonconforming FEM vs nonconforming meshes (clarified). $\endgroup$
    – Jesse Chan
    Commented Dec 1, 2015 at 18:31

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