Orthogonality constrained gradient reconstruction for superconvergent linear functionals

Abstract

The post-processing of the solution of variational problems discretized with Galerkin finite element methods is particularly useful for the computation of quantities of interest. Such quantities are generally expressed as linear functionals of the solution and the error of their approximation is bounded by the error of the solution itself. Several a posteriori recovery procedures have been developed over the years to improve the accuracy of post-processed results. Nonetheless such recovery methods usually deteriorate the convergence properties of linear functionals of the solution and, as a consequence, of the quantities of interest as well. The paper develops an enhanced gradient recovery scheme able to both preserve the good qualities of the recovered gradient and increase the accuracy and the convergence rates of linear functionals of the solution.

Appendix

1.1 Problem data for the numerical tests

In Tables 6 we summarize the problem data for the test cases of Sect. 5.

Table 6 Problem parameters for the monodimensional (1D), bidimensional (2D) and tridimensional (3D) test cases

1.2 Implementation of the SPR method

In Sect. 3 we have introduced the reconstructed gradient which is built on the Lagrangian tensorial basis functions $\{\varPsi _l\}_{l=1}^{L}$ having maximum degree identical to the one used for the approximated solution with nodal values $\{\zeta _l\}_{l=1}^{L}$:

(A.1)

The integer number L corresponds to the total number of degrees of freedom and is equal to $L=d \cdot C \cdot K$ where d is the dimension of the space, C represents the number of components of the unknown of the problem and K is the number of support points.

As illustrated in Fig. 1, to each support point it is associated at least one patch of cells ${\mathscr {P}}_k$ made of $2^d$ cells, such that . For each support point there are $d\cdot C$ components of to reconstruct. Each m-th component of the reconstructed gradient, with $m \in \{1,\ldots ,d\cdot C\}$, is then approximated in a least squares sense by a complete polynomial centered in of maximum degree p coincident with the degree used for the basis functions $\{\varPsi _l\}_{l=1}^{L}$. Hence the polynomial approximation on the patch ${\mathscr {P}}_k$ writes:

where $\mathrm{i}_k$ is an interval of $d\cdot C$ indices related to the index k, that is $\mathrm{i}_k :=\{d\cdot C\cdot (k-1)+1, \ldots , d\cdot C\cdot k\}$, represents the vector of the unknown coefficients and is the vector containing the polynomial basis functions. Then the nodal value $\zeta _l$ is set equal to the evaluation of the polynomial approximant at the recovery point:

(A.2)

The integer number B coincides with the number of Barlow points belonging to the current Patch ${\mathscr {P}}_k$, computed as:

$$\begin{aligned} B = \frac{\prod _{i=1}^{d}(p+i)}{d!}. \end{aligned}$$

For any component the coefficients vector is computed fitting the values of the gradient at the B Barlow points inside the patch ${\mathscr {P}}_k$ associated to , by resolution of a discrete least squares problem, which writes:

(A.3)

where $m = (l-1) \, \text {mod} \, (d\cdot C) + 1$. The operators $\text {div}$ and $\text {mod}$ represent respectively the integer division and the modulo operation.

The stationary point can be found by differentiation with respect to the minimization parameter . Setting the first-order derivative equal to zero we get the following linear algebraic system:

(A.4)

where the index k is related to the index l by the identity $k = (l-1) \, \text {div} \, (d\cdot C) + 1$ and the matrix $M_k$ and the right hand side are defined as:

(A.5)

Hence for each support point we solve $d\cdot C$ small linear problems (A.4) and we compute each nodal value $\zeta _l$ according to (A.2).

1.3 Implementation of the $\hbox {SPR}^+$ method

The constrained least squares problem that we want to solve has been discussed in Sect. 4 and derives from the SPR problem (A.6). We firstly rewrite the a-orthogonality condition as:

where and , as defined in Sect. 2. For any support point and its associated patch ${\mathscr {P}}_k$, saying that we want to reconstruct the m-th component , the constrained discrete least squares problem writes:

(A.6)

where are the Barlow points belonging to the patch ${\mathscr {P}}_k$. We adopt the Lagrange multiplier method, then let us define the multiplier and the objective function of the problem as:

(A.7)

We recall that are the Barlow points belonging to the patch ${\mathscr {P}}_k$ associated to the recovery point and the index k is related to the index l according to the rule $k = (l-1) \, \text {div}\, (C\cdot d) + 1$. The index $m \in \{1, \ldots , d\cdot C\}$ refers to the component of the gradient that we want to reconstruct and is given by $m = (l-1) \, \text {mod} \, (d\cdot C) + 1$.

The stationary point can be found by imposing the first order derivatives to be equal to zero. Differentiation by the coefficients vector leads to:

(A.8)

where the matrix $M_k$ and the vector have already been defined in (A.5). Then we have:

On the other hand differentiation by the Lagrange multiplier $\lambda$ leads to the recovery of the constraint equation of problem (A.6), which writes:

Substituting the explicit expression for provided by result (A.8), we finally get the following solution for the Lagrange multiplier: