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# Flux balanced approximation with least-squares gradient for diffusion equation on polyhedral mesh

• * Corresponding author: Peter Frolkovič

The first and second author are supported by grants VEGA 1/0709/19 and 1/0436/20 and APVV-15-0522

• A numerical method for solving diffusion problems on polyhedral meshes is presented. It is based on a finite volume approximation with the degrees of freedom located in the centers of computational cells. A numerical gradient is defined by a least-squares minimization for each cell, where we suggest a restricted form in the case of discontinuous diffusion coefficient. The flux balanced approximation is proposed without numerically computing the gradient itself at the faces of computational cells in order to find a normal diffusive flux. To apply the method for parallel computations with a 1-ring neighborhood, we use an iterative method to solve the obtained system of algebraic equations. Several numerical examples illustrate some advantages of the proposed method.

Mathematics Subject Classification: Primary: 65N08; Secondary: 65M08.

 Citation: • • Figure 1.  An illustration of the basic notation used in the flux approximation

Figure 2.  The cubic domain and polyhedral mesh with average discretization size $\ell = .19$ used in the Sections 4.1 and 4.4

Figure 3.  The domain used for the examples in the Section 4.2. The left picture shows a cut for which the mesh is visible. The blue and red parts visualize the subdomains with the different constant diffusion coefficients. The right picture shows the outer boundary with the corresponding surface mesh

Figure 4.  The cut (left) and the surface (right) of the complex domain used for the example in the Section 4.3

Figure 5.  The three surfaces of the boundary of computational domain (left) and the isosurfaces of numerical solution (right) for the example in the Section 4.3

Table 1.  The error norms and EOCs for the example in the Section 4.1

 $\ell$ $K$ $E_2$ EOC $G_1$ EOC 1.90e-1 10 3.69e-3 7.97e-2 9.52e-2 10 1.17e-3 1.66 3.22e-2 1.31 4.76e-2 9 3.27e-4 1.85 1.41e-2 1.19 2.48e-2 9 7.53e-5 2.26 6.08e-3 1.30

Table 2.  The error norms and the EOCs for the example in the concentric spherical domain in the Section 4.2 for the flux balanced method. The full least square gradient approximation is presented in the columns from $2$ to $6$, and the restricted one in the columns from $7$ to $11$

 $\ell$ $K$ $E_2$ EOC $G_1$ EOC $K$ $E_2$ EOC $G_1$ EOC .121 8 6.43e-5 3.41e-3 8 6.43e-5 1.07e-3 .090 8 3.08e-5 2.42 2.26e-3 1.36 8 3.08e-5 2.42 5.79e-4 2.03 .072 8 1.80e-5 2.44 1.70e-3 1.29 8 1.80e-5 2.44 3.93e-4 1.75 .060 8 1.15e-5 2.46 1.34e-3 1.27 8 1.15e-5 2.46 2.88e-4 1.71

Table 3.  The error norms and the EOCs for the example in the concentric spherical domain in the Section 4.2 using the method based on the face gradient approximation. The full least square gradient approximation is presented in the columns from $2$ to $6$, and the restricted one in the columns from $7$ to $11$

 $\ell$ $K$ $E_2$ EOC $G_1$ EOC $K$ $E_2$ EOC $G_1$ EOC .121 10 2.99e-4 6.44e-3 10 2.83e-4 3.49e-3 .090 9 2.07e-4 1.21 4.53e-3 1.16 9 1.94e-4 1.23 2.49e-3 1.11 .072 9 1.59e-4 1.20 3.48e-3 1.18 9 1.48e-4 1.22 1.92e-3 1.17 .060 9 1.28e-4 1.16 2.81e-3 1.16 9 1.19e-4 1.19 1.56e-3 1.16

Table 4.  The error norms and the EOCs for the example with Perona-Malik equation in the Section 4.4

 $\ell$ $K$ $E_2$ EOC $G_1$ EOC 1.90e-1 7 1.84e-4 1.70e-2 9.52e-2 6 5.31e-5 1.79 6.66e-3 1.36 4.76e-2 6 1.55e-5 1.78 2.96e-3 1.17 2.48e-2 5 4.03e-6 2.07 1.29e-3 1.28
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