Lines Matching +full:- +full:t

3 (example-petsc-area)=
10 …-Volume`, with the exception that the physical coordinates for this problem are $\bm{x}=(x,y,z)\in…
12 (example-petsc-area-cube)=
16 This is one of the test cases of the computation of the {ref}`example-petsc-area` of a 2D manifold …
20 ./area -problem cube
27 $$ (eq-coordinate-transforms-cube)
33 $$ (eq-jacobian-cube)
35 We note that in equation {eq}`eq-coordinate-transforms-cube`, the right-most Jacobian matrix ${\par…
41 (example-petsc-area-sphere)=
45 This problem computes the surface {ref}`example-petsc-area` of a tensor-product discrete sphere, ob…
46 …e is sometimes referred to as a cubed-sphere (an example of such as a surface is given in figure {…
50 ./area -problem sphere
53 (fig-cubed-sphere)=
56 Example of a cubed-sphere, i.e., a tensor-product discrete sphere, obtained by
60 … as coordinates on a circle and 1D linear element in figure {numref}`fig-sphere-coords`), to $\bm{…
64 $$ (eq-coordinate-transforms-sphere)
70 $$ (eq-jacobian-sphere)
72 (fig-sphere-coords)=
80 We note that in equation {eq}`eq-coordinate-transforms-sphere`, the right-most Jacobian matrix ${\p…
90 …= \frac{1}{\lVert \bm x \rVert} \bm I_{(3\times 3)} - \frac{1}{\lVert \bm x \rVert^3} (\bm x \bm x…
93 (example-petsc-bps)=
98 A generalization of these BPs to the surface of the cubed-sphere are available in `bpssphere.c`.
100 (example-petsc-bps-sphere)=
102 ### Bakeoff problems on the cubed-sphere
104 …-BP2, that use the mass operator, the coordinate transformations and the corresponding Jacobian de…
105 …-BP6, on the cubed-sphere, in addition to equation {eq}`eq-jacobian-sphere`, the pseudo-inverse of…
106 We begin by expressing the Moore-Penrose (left) pseudo-inverse:
109 …T \frac{\partial\overset{\circ}{\bm{x}}}{\partial \bm{X}}_{(3\times2)} \right)^{-1} \frac{\partial…
110 $$ (eq-dxcircdX-pseudo-inv)
121 …rtial \overset\circ{\bm x}} \left( \frac{\partial u}{\partial \overset\circ{\bm x}} \right)^T \, dS
122 …\partial \overset\circ{\bm x}} \right)^T}_{\bm g_{(2\times 2)}}  \left(\frac{\partial u}{\partial …
123 $$ (eq-weak-laplace-sphere)
125 …epresents the surface of the sphere, which is a two-dimensional closed surface embedded in the thr…
126 This expression can be simplified to avoid the explicit Moore-Penrose pseudo-inverse,
129 …T \frac{\partial\overset{\circ}{\bm{x}}}{\partial \bm{X}} \right)^{-1}_{(2\times 2)} \frac{\partia…
133 This allows us to simplify {eq}`eq-weak-laplace-sphere` as
136 …T \, dS     = \int_{\Omega} \frac{\partial v}{\partial \bm X} \underbrace{\left(\frac{\partial \ov…
141 (example-petsc-multigrid)=
146 It investigates $p$-multigrid for the Poisson problem, equation {eq}`eq-variable-coeff-poisson`, us…
150 -\nabla\cdot \left( \kappa \left( x \right) \nabla x \right) = g \left( x \right)
151 $$ (eq-variable-coeff-poisson)
153 …e {ref}`fig-operator-decomp`, and the restriction and prolongation operators given by interpolatio…
154 These three operations can be exploited by existing matrix-free multigrid software and smoothers.