StartFraction h Superscript k minus 1 Baseline Over p Superscript k minus 1 Baseline EndFraction double-vertical-bar u Subscript upper E upper X Baseline double-vertical-bar Subscript upper H Sub Superscript k Subscript left-parenthesis normal upper Omega right-parenthesis Baseline 2nd Column for k minus 1 less-than-or-equal-to p 2nd Row 1st Column Blank 2nd Column Blank 3rd Row 1st Column upper C left-parenthesis k right-parenthesis StartFraction h Superscript p Baseline Over p Superscript k minus 1 Baseline EndFraction double-vertical-bar u Subscript upper E upper X Baseline double-vertical-bar Subscript upper H Sub Superscript p plus 1 Subscript left-parenthesis normal upper Omega right-parenthesis Baseline 2nd Column for k minus 1 greater-than p EndMatrix"/>
where is the energy norm, k is typically a fractional number and is a positive constant that depends on k but not on h or p. This inequality gives the upper bound for the asymptotic rate of convergence of the relative error in energy norm as or [22]. This estimate holds for one, two and three dimensions. For one and two dimensions lower bounds were proven in [13, 24] and [46] and it was shown that when singularities are located in vertex points then the rate of convergence of the p‐version is twice the rate of convergence of the h‐version when both are expressed in terms of the number of degrees of freedom. It is reasonable to assume that analogous results can be proven for three dimensions; however, no proofs are available at present.
We will find it convenient to write the relative error in energy norm in the following form
(1.92)
where N is the number of degrees of freedom and C and β are positive constants, β is called the algebraic rate of convergence. In one dimension for the h‐version and for the p‐version. Therefore for we have . However, for the important special case when the solution has the functional form of eq. (1.89) or, more generally, has a term like and is a nodal point then for the p‐version: The rate of p‐convergence is twice that of h‐convergence [22, 84].
When the exact solution is an analytic function then and the asymptotic rate of convergence is exponential:
(1.93)
where C, γ and θ are positive constants, independent of N. In one dimension , in two dimensions , in three dimensions , see [10].
When the exact solution is a piecewise analytic function then eq. (1.93) still holds provided that the boundary points of analytic functions are nodal points, or more generally, lie on the boundaries of finite elements.
The relationship between the error measured in energy norm and the error in potential energy is established by the following theorem.
Theorem 1.5
(1.94)
Proof: Writing and noting that , from the definition of we have:
Remark 1.10 Consider the problem given by eq. (1.5) and assume that κ and c are constants. In this case the smoothness of u depends only on the smoothness of f: If
