Results
The Laplace equation
The accuracy of FEM approximation of the Laplace equation on different meshes were examined. Numerical results vs. analytic ones for cubic and spherical domain are presented in Fig.4. The relative difference between both analytic and numerical solutions has been calculated as
|
(φnum - φexact)/max(φexact).
|
The Laplace equation has been solved for the cubic domain [0, π] ✗ [0, π] ✗ [0, π] with potential function φ = 0 everywhere on the boundary Γ apart from one its side at x = π where potential φ = 1 and for the spherical domain with the boundary conditions imposed by putting an elementary charge outside the sphere in [0, 0, 2π]. The exact solutions for both considered cases are evaluated precisely in1. FEM approximation has been computed for the ,,linear'' order of tetrahedron2. However, the comparison between both orders of approximation i .e ,,linear'' and ,,quadratic'' for the uniform mesh with Velem = 0.0202 has been performed. The formulas for higher orders of approximation i.e. quadratic and cubic can be found in2. The results show that mean discrepancy between numerical and analytic solutions calculated according to equation for the Laplace equation equal -0.0061 ± 0.0153 in the linear case and 0.0004 ± 0.0082 in quadratic approximation, respectively. However, in further studies linear approximation will be used as being sufficiently accurate. In the case of cubic domain a mesh of unique element volume (non-optimized) has been applied in contrast to the spherical domain where mesh has been used after its enhancement with both the Metropolis algorithm and the Delaunay routine.
Fig. 4. The picture shows a) FEM approximation (linear) of the Laplace problem at the center of the cubic domain i.e. at z = π/2 (as described in) vs. the analytic result together with b) a distribution of differences between numerical and analytic solutions obtained for each node in Ω domain; c) FEM approximation (linear) vs. the exact solution of the Laplace equation for the spherical domain with the boundary values defined by putting an elementary charge outside the sphere in [0, 0, 2π] b) the volume distribution in the spherical domain optimized with the Metropolis algorithm with elements of a prescribed volume V0 equals 0.0075.
References
