## 5 Numerical Tests
### Solution Convergence of the Problems  
In the convergence studies, the logarithm of the approximation error in potential energy is presented as a function of the logarithm of the number of dofs \(N\) of the problem. The corresponding curves are \(p\)-convergence ones as the number of dofs in the problems increase due to the change of the longitudinal order of approximation \(p\) within elements. In order to present the solution convergence for the three model problems with three types of the transition elements applied, three threesomes of pictures are displayed. The first three pictures (Figures 20-22) correspond to the bending-dominated plate problem, the second threesome (Figures 23-25) to the bending-dominated shell problem, while the last three drawings (Figures 26-28) to the membrane-dominated shell. In the figures, the first, second and third drawings correspond to the cases of the classical, modified, and enhanced transition elements applied. The mentioned approximation error is defined as the difference of two energy measures of which \(E\) corresponds to the numerical solution under consideration and the reference solution \(E_{r}\) plays a role of the exact solution, namely:  
\[energy\ error=|E_{r}-E|. \tag{84}\]  
The energy measures represent the sum of absolute values of the mechanical and electrical parts of the electro-mechanical potential energy, i.e.,  
\[E=|B(\mathbf{u},\mathbf{u})-C(\mathbf{u},\phi)|+|-b(\phi,\phi)-C(\phi,\mathbf{u})|. \tag{85}\]  
Due to stationarity of the solutions \(\mathbf{u}\) and \(\phi\), resulting from (8), the potential energy can be expressed with the components of the electro-mechanical co-energy (strain, electric field, and coupling energies) corresponding to the bilinear forms \(B\), \(b\) and \(C\) from (9)-(11). The solutions for displacements \(\mathbf{u}\) and electric potential \(\phi\) are approximated with use of the global nodal vectors of the displacement dofs \(\mathbf{q}^{hpq}\) and potential dofs \(\mathbf{q}^{hmp}\) obtained from solution of the problem (35). Figure 20: Convergence curves—the plate problem, the classical transition elements employed. Figure 21: Convergence curves—the plate problem, the modified transition elements applied. Figure 22: Convergence curves—the plate problem, the enhanced transition elements applied. Figure 23: Convergence curves—the bending-dominated shell with the classical transition elements. Figure 24: Convergence curves—the bending-dominated shell with the modified transition elements. Figure 26: Convergence curves—the membrane shell with the classical transition elements. Figure 25: Convergence curves—the bending-dominated shell with the enhanced transition elements. The proposed measure of the approximation error was applied as it guarantees monotonicity of the energy approximation error with changing value of \(p\equiv\pi\), while the simple sum of both parts of the energy (equal to the electro-mechanical co-energy) may not. Note that such monotonicity is a must in the case of error-controlled adaptivity suggested in Reference [11; 12]. Figure 28: Convergence curves—the membrane shell with the enhanced transition elements. \[E=|B(\mathbf{u},\mathbf{u})-C(\mathbf{u},\phi)|+|-b(\phi,\phi)-C(\phi,\mathbf{u})|. \tag{85}\]0 Convergence curves—the membrane shell with the modified transition elements. In order to calculate the errors, the exact values \(E_{r}\) of the energy measures are replaced with their best numerical approximations obtained from the so-called over-killed meshes, i.e., the meshes for which the longitudinal order of approximation is equal to \(p=\pi=9\). In the case of two bending-dominated problems the convergence for the enhanced models is slightly higher than for the modified transition elements employed in the mixed models. Additionally, the latter elements produce slightly better convergence than the classical ones. In the case of the membrane-dominated problem, only the enhanced transition elements deliver convergence curves between the curves obtained for the basic models.