Figure 6 - uploaded by Robert L Gates
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The problem of solving partial differential equations (PDEs) on manifolds can be considered to be one of the most general problem formulations encountered in computational multi-physics. The required covariant forms of balance laws as well as the corresponding covariant forms of the constitutive closing relations are naturally expressed using the b...
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... verification of our implementation without adaptivity, we compute the kernel of the Hodge Laplacian problem with k = n − 1 = 1 on the torus with Betti number b 1 = 2 embedded in R 3 . Using lowest-order elements of type σ ∈ P − 1 Λ 0 and u ∈ P − 1 Λ 1 we thus obtain an approximation of the basis for the harmonic forms H 1 , which is shown in Fig. 6. It can clearly be seen that the covector fields obtained circulate around the 1-holes of the torus. To demonstrate the potential of adaptivity, we solve the Hodge Laplacian problem for k = n = 2, i.e. the mixed formulation of the scalar Laplacian, on an L-shaped domain. By Arnold, Falk and Winther (2006) [3], we choose as a stable ...
Context 2
... verification of our implementation without adaptivity, we compute the kernel of the Hodge Laplacian problem with k = n − 1 = 1 on the torus with Betti number b 1 = 2 embedded in R 3 . Using lowest-order elements of type σ ∈ P − 1 Λ 0 and u ∈ P − 1 Λ 1 we thus obtain an approximation of the basis for the harmonic forms H 1 , which is shown in Fig. 6. It can clearly be seen that the covector fields obtained circulate around the 1-holes of the ...
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