Matthew Knepley <[email protected]> writes: >> Unfortunately, something more is required for higher order accuracy, >> since naively the coordinate section itself would have to be higher >> order, and this would require lots of changes (the equivalent of >> DMPlexComputeCellGeometry would be called once per quadrature point >> instead of once per element). >> > > I have never been convinced that isoperimetric stuff produces enough > benefit for its complication. Polynomials are not good approximators > for the Jacobian of these transforms. NURBS are so much better.
These issues are orthogonal. If the mapping is not affine, you need separate Jacobians at each quadrature point. Non-affine elements are required for high-order accuracy with curved boundaries, and in many cases when using quad and hex elements. The value of NURBS is that (a) some coordinate transformations can be represented exactly and (b) for certain problems, the rest solution can be represented exactly in the ansatz space. Quadrature error does not magically vanish.
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