Reference elements and affine maps

Let the physical simplex K be the image of a fixed reference simplex under x=B_K xi+b_K, with an invertible matrix B_K. Define the pulled-back scalar function by evaluating the physical function at that image. The chain rule gives the reference gradient as B_K transpose times the physical gradient. Solving for the physical gradient produces the inverse transpose. Translation changes locations but does not enter the derivative matrix; using B_K instead of its inverse transpose generally gives the wrong directional scale.

The change of variables for volume uses the absolute determinant, because integration measure does not depend on orientation. A negative determinant may reflect a reversed vertex ordering rather than a negative geometric area. Scalar Lagrange elements use composition with the map; vector-valued H(div) or H(curl) elements require appropriate Piola transformations to preserve normal or tangential quantities. The scalar formulas here should not be extended to those spaces without changing the pullback.

Insert the gradient and measure transformations into the unit-diffusion energy integral. The transformed integrand contains the matrix B_K inverse times its transpose inverse. This matrix records directional distortion. For a uniform scaling by h in d dimensions, volume scales as h to the d and two derivatives contribute h to the negative two, giving stiffness scale h to the d minus two. A pure scalar volume multiplier would miss both inverse derivative factors.

The element stiffness is positive semidefinite before essential boundary constraints because constants have zero gradient. Local row sums therefore vanish for a basis reproducing constants. Global assembly combines shared degrees of freedom and imposes the boundary constraints that remove the relevant null modes. These checks help detect transformation errors, but they do not replace correct quadrature or a valid mesh. A positive element area alone is not sufficient evidence that every directional derivative has been handled correctly.

The cell diameter alone does not control the inverse map. A long thin triangle can have a moderate diameter and a tiny altitude, giving a large inverse derivative factor in its thin direction. A shape-regular simplicial family bounds the ratio of diameter to inradius uniformly. Relative to a fixed reference simplex, this controls the condition of the normalized affine maps and yields comparable upper and lower singular-value scales proportional to the local diameter.

Shape regularity permits different cell sizes across a mesh; it is weaker than quasi-uniformity, which also compares the largest and smallest sizes globally. Local interpolation estimates can use local diameters on shape-regular graded meshes without assuming every cell has the same size. Anisotropic meshes can be useful, but their analysis needs directional estimates rather than blindly applying an isotropic constant that is no longer uniform. Record the geometry assumptions before interpreting an observed refinement rate as a theorem.