In modified form taken from

Ralf Hartmann
Adaptive Finite Element Methods for the Compressible Euler Equations
PhD thesis, University of Heidelberg, 2002.

Higher order Boundary approximation

Introduction: In many numerical applications the domain , is not a polygonal domain but includes curved boundaries. For these cases the boundary cannot be represented exactly by the discretised boundary. Approximating the boundary by a piecewise linear boundary interpolation, i.e. by a polygonal boundary, may in some applications not be sufficient. In these cases a higher order boundary approximation, for example by piecewise quadratic or cubic boundary interpolation, must be employed. In the finite element framework this higher order boundary approximation is realized by mapping the reference element to the element in real space, whereas on cells K at the boundary, i.e. , the mappings are given by polynomial functions of higher degree.

Elements with general mapping functions . We begin by first introducing some notation. Let be a cell of the triangulation Th with , where is a smooth bijective mapping of the reference element (unit square) to the element K in real space, see Figure 1.

In the following and for the sake of simplicity we suppress the letter K in the subscript and write instead of .

Mapping functions of higher polynomial degree. A mapping function , that maps the reference element to an arbitrary quadrilateral cell K with straight boundaries, can in general be represented by a bilinear function, i.e. by a Q1-mapping. For the case that the cell K includes curved boundaries it might be necessary to employ polynomial mapping functions of higher degree.

Given a degree p>0, a cell , and (p+1)d mapping support points , , we define a Qp-mapping as follows

 (1)

Here, , denote the Lagrange interpolation basis functions, that satisfy

where , denote the Lagrange support points on the unit cell . The definition of (1) ensures that each of the unit support points is mapped onto the corresponding mapping support points pi, i.e.

 (2)

Analogous to Lagrange finite elements the unit Lagrange support points are equidistantly distributed on based on a tensor product mesh. In the following we only consider the two-dimensional case, d=2. For that case, Figure 2 shows the distributions of the unit support points , for degrees .

Let the ordering and numbering of the unit support points be as follows: first the corners, then the points on the edges and finally the inner support points, see also Figure 2. Thus the first 4p points are placed on the boundary of the reference cell, i.e.

According to (2) these points are mapped to the mapping support points pk, that are chosen to be placed on the boundary of the real cell in approximatively equal distances, i.e.

While the support points pk, on the boundary are given by the boundary description of the real cell K, the inner mapping support points

are not uniquely determined. Numerical tests show that it is not a trivial task to define the positions of the inner mapping support points appropriately. If they are not chosen appropriately the resulting mapping for a cell K may degenerate, i.e. the mapping for some cell K may not be bijective.

Computation of inner support points by smooth transformation. In the following we will define the positions of the inner mapping support points so that the mapping does - in all practical cases - not degenerate. To this end, we employ an approach for the mapping of the support points, that is in the style of the smooth transformations that is used to transform structured triangulations to match complex boundary discriptions. In the following, again for notational convenience, we consider only the two-dimensional case.

The smooth transformation mentioned above is based on solutions to the Laplace equation that is solved on the reference cell . Discrete boundary conditions are imposed that are given by the coordinates of the mapping support points pk, , on the boundary of the cell K in real space.

To be more explicite we define a Laplace problem on the unit cell

 (3)

for each component , l=1,2, of the Qp mapping . Here, the discrete boundary function is given by

 (4)

where (pi)l denotes the lth component of the support point pi, and the corresponding Lagrangian interpolation basis function. We recall that the numbering of the mapping support points involves for . Substituting

 (5)

into the Laplace problem (3) yields the zero boundary value problem,

 (6)

that is equivalent to the following variational formulation

Discretisation of this problem

and recalling definitions (1), (5) and (4) gives

 (7)

with the matrices and given by

and

The solutions to problem (7) for l=1,2 are

that may be written in compact form:

 (8)

where cjk represents the coefficient

 (9)

of the linear combination (8), that represents the dependency of the jth inner mapping support point p4p+j on the support points pk, , that are placed on the boundary of the cell K. For a fixed degree p, these coefficients cjk are the same for the mapping of all cells K in real space because the cjk depend only on the reference element . Therefore the coefficients cjk can be precomputed and result in following linear combinations:

For p=2 the linear combination turns out to be