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Journal of Computational and Theoretical Transport Volume 47, 2018 - Issue 1-3: Proceedings of the 25th International Conference on Transport Theory, Part I: Transport Theory

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A Priori Error Estimates and Computational Studies for a Fermi Pencil-Beam Equation

M. AsadzadehDepartment of Mathematical Sciences, Chalmers University of Technology and Gothenburg University, Gothenburg, SwedenCorrespondence[email protected]
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L. BeilinaDepartment of Mathematical Sciences, Chalmers University of Technology and Gothenburg University, Gothenburg, SwedenView further author information

M. NaseerDepartment of Mathematical Sciences, Chalmers University of Technology and Gothenburg University, Gothenburg, SwedenView further author information

C. StandarDepartment of Mathematical Sciences, Chalmers University of Technology and Gothenburg University, Gothenburg, SwedenView further author information

Pages 125-151 | Published online: 20 Feb 2019

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https://doi.org/10.1080/23324309.2018.1496937
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Figure 1. Test 1(a). (a)–(d) Locally adaptively refined meshes of . (e) Computed solution on the four times adaptively refined mesh (d).

Figure 1. Test 1(a). (a)–(d) Locally adaptively refined meshes of Table 1. (e) Computed solution on the four times adaptively refined mesh (d).

Figure 2. Test 1(b). (a)–(d) Locally adaptively refined meshes of . (e) Computed solution on the four times adaptively refined mesh (d).

Figure 2. Test 1(b). (a)–(d) Locally adaptively refined meshes of Table 2. (e) Computed solution on the four times adaptively refined mesh (d).

Table 1. Test 1(a). Computed errors $e_{n} = | | u - u_{h}^{n} | |_{L_{2} (Ω_{⊥})}$ and $e_{n - 1} / e_{n}$ on the adaptively refined meshes. Here, the solution $u_{h}^{n}$ is computed using semi-streamline diffusion method of Section 4.1 with $\tilde{γ} = 0.5$ in the adaptive algorithm and $α = 0.1$ in (86).

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Table 2. Test 1(b). Computed errors $e_{n} = | | u - u_{h}^{n} | |_{L_{2} (Ω_{⊥})}$ and $e_{n - 1} / e_{n}$ on the adaptively refined meshes. Here, the solution $u_{h}^{n}$ is computed using semi-streamline diffusion method of Section 4.1 with $\tilde{γ} = 0.7$ in the adaptive algorithm and $α = 0.1$ in (86).

Display Table

Figure 3. Test 2(a). (a)–(d) Locally adaptively refined meshes of . (e) Computed solution on the four times adaptively refined mesh (d).

Figure 3. Test 2(a). (a)–(d) Locally adaptively refined meshes of Table 3. (e) Computed solution on the four times adaptively refined mesh (d).

Figure 4. Test 2(b). (a)–(d) Locally adaptively refined meshes of . (e) Computed solution on the four times adaptively refined mesh (d).

Figure 4. Test 2(b). (a)–(d) Locally adaptively refined meshes of Table 4. (e) Computed solution on the four times adaptively refined mesh (d).

Table 3. Test 2(a). Computed values of errors $e_{n} = | | u - u_{h}^{n} | |_{L_{2} (Ω_{⊥})}$ and $e_{n - 1} / e_{n}$ on the adaptively refined meshes. Here, the solution $u_{h}^{n}$ is computed using characteristic streamline diffusion method with $\tilde{γ} = 0.5$ in the adaptive algorithm.

Display Table

Table 4. Test 2(b). Computed values of errors $e_{n} = | | u - u_{h}^{n} | |_{L_{2} (Ω_{⊥})}$ and $e_{n - 1} / e_{n}$ on the adaptively refined meshes. Here, the solution $u_{h}^{n}$ is computed using characteristic streamline diffusion method with $\tilde{γ} = 0.7$ in the adaptive algorithm.

Display Table

Figure 5. Test 3(a). (a)–(d) Locally adaptively refined meshes of . (e) Computed solution on the four times adaptively refined mesh (d).

Figure 5. Test 3(a). (a)–(d) Locally adaptively refined meshes of Table 5. (e) Computed solution on the four times adaptively refined mesh (d).

Figure 6. Test 3(b). (a)–(d) Locally adaptively refined meshes of . (e) Computed solution on the four times adaptively refined mesh (d).

Figure 6. Test 3(b). (a)–(d) Locally adaptively refined meshes of Table 6. (e) Computed solution on the four times adaptively refined mesh (d).

Table 5. Test 3(a). Computed values of errors $e_{n} = | | u - u_{h}^{n} | |_{L_{2} (Ω_{⊥})}$ and $e_{n - 1} / e_{n}$ on the adaptively refined meshes. Here, the solution $u_{h}^{n}$ is computed using semi-streamline diffusion method with $\tilde{γ} = 0.5$ in the adaptive algorithm.

Display Table

Table 6. Test 3(b). Computed values of errors $e_{n} = | | u - u_{h}^{n} | |_{L_{2} (Ω_{⊥})}$ and $e_{n - 1} / e_{n}$ on the adaptively refined meshes. Here, the solution $u_{h}^{n}$ is computed using semi-streamline diffusion method with $\tilde{γ} = 0.7$ in the adaptive algorithm.

Display Table

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A Priori Error Estimates and Computational Studies for a Fermi Pencil-Beam Equation

Table 3. Test 2(a). Computed values of errors $e_{n} = | | u - u_{h}^{n} | |_{L_{2} (Ω_{⊥})}$ and $e_{n - 1} / e_{n}$ on the adaptively refined meshes. Here, the solution $u_{h}^{n}$ is computed using characteristic streamline diffusion method with $\tilde{γ} = 0.5$ in the adaptive algorithm.

Table 4. Test 2(b). Computed values of errors $e_{n} = | | u - u_{h}^{n} | |_{L_{2} (Ω_{⊥})}$ and $e_{n - 1} / e_{n}$ on the adaptively refined meshes. Here, the solution $u_{h}^{n}$ is computed using characteristic streamline diffusion method with $\tilde{γ} = 0.7$ in the adaptive algorithm.

Table 5. Test 3(a). Computed values of errors $e_{n} = | | u - u_{h}^{n} | |_{L_{2} (Ω_{⊥})}$ and $e_{n - 1} / e_{n}$ on the adaptively refined meshes. Here, the solution $u_{h}^{n}$ is computed using semi-streamline diffusion method with $\tilde{γ} = 0.5$ in the adaptive algorithm.

Table 6. Test 3(b). Computed values of errors $e_{n} = | | u - u_{h}^{n} | |_{L_{2} (Ω_{⊥})}$ and $e_{n - 1} / e_{n}$ on the adaptively refined meshes. Here, the solution $u_{h}^{n}$ is computed using semi-streamline diffusion method with $\tilde{γ} = 0.7$ in the adaptive algorithm.

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A Priori Error Estimates and Computational Studies for a Fermi Pencil-Beam Equation

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Table 1. Test 1(a). Computed errors en=||u−uhn||L2(Ω⊥) and en−1/en on the adaptively refined meshes. Here, the solution uhn is computed using semi-streamline diffusion method of Section 4.1 with γ˜=0.5 in the adaptive algorithm and α=0.1 in (86).

Table 2. Test 1(b). Computed errors en=||u−uhn||L2(Ω⊥) and en−1/en on the adaptively refined meshes. Here, the solution uhn is computed using semi-streamline diffusion method of Section 4.1 with γ˜=0.7 in the adaptive algorithm and α=0.1 in (86).

Table 3. Test 2(a). Computed values of errors en=||u−uhn||L2(Ω⊥) and en−1/en on the adaptively refined meshes. Here, the solution uhn is computed using characteristic streamline diffusion method with γ˜=0.5 in the adaptive algorithm.

Table 4. Test 2(b). Computed values of errors en=||u−uhn||L2(Ω⊥) and en−1/en on the adaptively refined meshes. Here, the solution uhn is computed using characteristic streamline diffusion method with γ˜=0.7 in the adaptive algorithm.

Table 5. Test 3(a). Computed values of errors en=||u−uhn||L2(Ω⊥) and en−1/en on the adaptively refined meshes. Here, the solution uhn is computed using semi-streamline diffusion method with γ˜=0.5 in the adaptive algorithm.

Table 6. Test 3(b). Computed values of errors en=||u−uhn||L2(Ω⊥) and en−1/en on the adaptively refined meshes. Here, the solution uhn is computed using semi-streamline diffusion method with γ˜=0.7 in the adaptive algorithm.

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Table 3. Test 2(a). Computed values of errors $e_{n} = | | u - u_{h}^{n} | |_{L_{2} (Ω_{⊥})}$ and $e_{n - 1} / e_{n}$ on the adaptively refined meshes. Here, the solution $u_{h}^{n}$ is computed using characteristic streamline diffusion method with $\tilde{γ} = 0.5$ in the adaptive algorithm.

Table 4. Test 2(b). Computed values of errors $e_{n} = | | u - u_{h}^{n} | |_{L_{2} (Ω_{⊥})}$ and $e_{n - 1} / e_{n}$ on the adaptively refined meshes. Here, the solution $u_{h}^{n}$ is computed using characteristic streamline diffusion method with $\tilde{γ} = 0.7$ in the adaptive algorithm.

Table 5. Test 3(a). Computed values of errors $e_{n} = | | u - u_{h}^{n} | |_{L_{2} (Ω_{⊥})}$ and $e_{n - 1} / e_{n}$ on the adaptively refined meshes. Here, the solution $u_{h}^{n}$ is computed using semi-streamline diffusion method with $\tilde{γ} = 0.5$ in the adaptive algorithm.

Table 6. Test 3(b). Computed values of errors $e_{n} = | | u - u_{h}^{n} | |_{L_{2} (Ω_{⊥})}$ and $e_{n - 1} / e_{n}$ on the adaptively refined meshes. Here, the solution $u_{h}^{n}$ is computed using semi-streamline diffusion method with $\tilde{γ} = 0.7$ in the adaptive algorithm.