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Fig. 7. Error norms decay for the numerical solution to problem (73) employing an unstructured mesh. Solid line: Lz norm; dashed line: L,, norm; dash-dot: L; norm.

Figure 7 Error norms decay for the numerical solution to problem (73) employing an unstructured mesh. Solid line: Lz norm; dashed line: L,, norm; dash-dot: L; norm.

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Fig. 1. Grid system in the transformed space for p = 2. Elements are bounded by solid lines, control volumes by dashed lines, interpolation points are indicated by dots. Shaded areas indicate control volumes belonging to one, two and four different elements.
Fig. 1. Grid system in the transformed space for p = 2. Elements are bounded by solid lines, control volumes by dashed lines, interpolation points are indicated by dots. Shaded areas indicate control volumes belonging to one, two and four different elements.
Fig. 2. Conditioning of the global coefficient matrix applied to the one-dimensional diffusion equation. Squares indicate uniform interpolation nodes and uniform control-volume edges. Circles indicate uniform interpolation nodes and Legendre control-volume edges. Triangles indicate Legendre interpolation nodes and uniform control-volume edges. Plus signs indicate Chebyshev interpolation nodes and Chebyshev control-volume edges. Diamonds indicate Legendre interpolation nodes and Legendre control-volume edges.
Fig. 2. Conditioning of the global coefficient matrix applied to the one-dimensional diffusion equation. Squares indicate uniform interpolation nodes and uniform control-volume edges. Circles indicate uniform interpolation nodes and Legendre control-volume edges. Triangles indicate Legendre interpolation nodes and uniform control-volume edges. Plus signs indicate Chebyshev interpolation nodes and Chebyshev control-volume edges. Diamonds indicate Legendre interpolation nodes and Legendre control-volume edges.
Error norms and order of convergence in the solution to problem (65) on a uniform mesh with n, elements. Fifth-order elements (n, = 6) are used.
Error norms and order of convergence in the solution to problem (65) on a uniform mesh with n, elements. Fifth-order elements (n, = 6) are used.
Fig. 3. Efficiency index E, computed with the L; error norm. Circles indicate the quadrature-based method and y =0; triangles: quadrature-free method and y = 0; squares: quadrature-based method and y=200f; diamonds: quadrature-free method and y = 2008.
Fig. 3. Efficiency index E, computed with the L; error norm. Circles indicate the quadrature-based method and y =0; triangles: quadrature-free method and y = 0; squares: quadrature-based method and y=200f; diamonds: quadrature-free method and y = 2008.
Error norms for the one-dimensional advection-diffusion problem for Pe = 100. The computational domain is subdivided into four elements. Table 2
Error norms for the one-dimensional advection-diffusion problem for Pe = 100. The computational domain is subdivided into four elements. Table 2
Fig. 4. Error norm evaluated for the numerical solution of the boundary value problem (65) on a single element. Squares indicate LL distribution, circles are for LU distribution; crosses for UL distribution while triangles mark the UU distribution.
Fig. 4. Error norm evaluated for the numerical solution of the boundary value problem (65) on a single element. Squares indicate LL distribution, circles are for LU distribution; crosses for UL distribution while triangles mark the UU distribution.
Fig. 5. Error norm L, for the solution of problem (67). Solid lines: p +2 (p+1 for p= 1) polynomial convergence rates. Dashed lines: p+ 1 and p+ 3 (p and p +2 for p=1) polynomial convergence rates. Symbols identify results computed with different polynomial degrees; circles, p=1; squares, p=2; triangles pointing downwards, p = 3; triangles pointing upwards, p = 4; x-symbols, p=5 . The following one-dimensional boundary value problem is solved numerically between 0 and 1.
Fig. 5. Error norm L, for the solution of problem (67). Solid lines: p +2 (p+1 for p= 1) polynomial convergence rates. Dashed lines: p+ 1 and p+ 3 (p and p +2 for p=1) polynomial convergence rates. Symbols identify results computed with different polynomial degrees; circles, p=1; squares, p=2; triangles pointing downwards, p = 3; triangles pointing upwards, p = 4; x-symbols, p=5 . The following one-dimensional boundary value problem is solved numerically between 0 and 1.
Order of accuracy evaluated for the numerical solution to problem (73) on Cartesian grids with three different arrangements. The order of accuracy is evaluated form the comparison of the L2 error norm on the grids reported in the first column. 4.4.2. Increase of the expansion order on the unstructured mesh
Order of accuracy evaluated for the numerical solution to problem (73) on Cartesian grids with three different arrangements. The order of accuracy is evaluated form the comparison of the L2 error norm on the grids reported in the first column. 4.4.2. Increase of the expansion order on the unstructured mesh
Fig. 6. Unstructured grid for the numerical solution to problem (73).
Fig. 6. Unstructured grid for the numerical solution to problem (73).
Fig. 8. First derivative along x of the analytical solution for the two-dimensional advection-diffusion problem (75). The same problem is solved using an unstructured mesh of stretched, skewed, distorted elements as shown in Fig. 10. The accuracy of the numerical solution is progressively improved through p-refinement. Fig. 11 shows the dependence of different error norms on the expansion order. An approximately linear
Fig. 8. First derivative along x of the analytical solution for the two-dimensional advection-diffusion problem (75). The same problem is solved using an unstructured mesh of stretched, skewed, distorted elements as shown in Fig. 10. The accuracy of the numerical solution is progressively improved through p-refinement. Fig. 11 shows the dependence of different error norms on the expansion order. An approximately linear
Fig. 11. Numerical error as a function of interpolation order between 1 and 17. Solid line: Lz norm; dashed line: L; norm; dash-dot: L,, norm.
Fig. 11. Numerical error as a function of interpolation order between 1 and 17. Solid line: Lz norm; dashed line: L; norm; dash-dot: L,, norm.
Fig. 9. Normalized error ¢/Ax?"? for the numerical solution of the two-dimensional advection-diffusion problem considered in Section 4.5. Filled circles indicate linear elements, squares are for p = 2, downward triangles for p = 3 upward triangles for p=4, stars for p=5.
Fig. 9. Normalized error ¢/Ax?"? for the numerical solution of the two-dimensional advection-diffusion problem considered in Section 4.5. Filled circles indicate linear elements, squares are for p = 2, downward triangles for p = 3 upward triangles for p=4, stars for p=5.
Fig. 10. Unstructured mesh used for the numerical solution of the two-dimensional advection-diffusion problem considered in Section 4.5. 5.4. Conclusions
Fig. 10. Unstructured mesh used for the numerical solution of the two-dimensional advection-diffusion problem considered in Section 4.5. 5.4. Conclusions
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