bps.md (cd3003229e5325bdf31f9e102d9fa328f6582fc1) bps.md (17be3a414c6fae47654f1361bae9c9dbcdd66795)
1(bps)=
2
3# CEED Bakeoff Problems
4
5```{include} ./README.md
6:start-after: bps-inclusion-marker
7:end-before: bps-exclusion-marker
8```
9
10(mass-operator)=
11
12## Mass Operator
13
1(bps)=
2
3# CEED Bakeoff Problems
4
5```{include} ./README.md
6:start-after: bps-inclusion-marker
7:end-before: bps-exclusion-marker
8```
9
10(mass-operator)=
11
12## Mass Operator
13
14The Mass Operator used in BP1 and BP2 is defined via the $L^2$ projection
15problem, posed as a weak form on a Hilbert space $V^p \subset H^1$, i.e.,
16find $u \in V^p$ such that for all $v \in V^p$
14The Mass Operator used in BP1 and BP2 is defined via the $L^2$ projection problem, posed as a weak form on a Hilbert space $V^p \subset H^1$, i.e., find $u \in V^p$ such that for all $v \in V^p$
17
18$$
19\langle v,u \rangle = \langle v,f \rangle ,
20$$ (eq-general-weak-form)
21
15
16$$
17\langle v,u \rangle = \langle v,f \rangle ,
18$$ (eq-general-weak-form)
19
22where $\langle v,u\rangle$ and $\langle v,f\rangle$ express the continuous
23bilinear and linear forms, respectively, defined on $V^p$, and, for sufficiently
24regular $u$, $v$, and $f$, we have:
20where $\langle v,u\rangle$ and $\langle v,f\rangle$ express the continuous bilinear and linear forms, respectively, defined on $V^p$, and, for sufficiently regular $u$, $v$, and $f$, we have:
25
26$$
27\begin{aligned} \langle v,u \rangle &:= \int_{\Omega} \, v \, u \, dV ,\\ \langle v,f \rangle &:= \int_{\Omega} \, v \, f \, dV . \end{aligned}
28$$
29
21
22$$
23\begin{aligned} \langle v,u \rangle &:= \int_{\Omega} \, v \, u \, dV ,\\ \langle v,f \rangle &:= \int_{\Omega} \, v \, f \, dV . \end{aligned}
24$$
25
30Following the standard finite/spectral element approach, we formally
31expand all functions in terms of basis functions, such as
26Following the standard finite/spectral element approach, we formally expand all functions in terms of basis functions, such as
32
33$$
34\begin{aligned}
35u(\bm x) &= \sum_{j=1}^n u_j \, \phi_j(\bm x) ,\\
36v(\bm x) &= \sum_{i=1}^n v_i \, \phi_i(\bm x) .
37\end{aligned}
38$$ (eq-nodal-values)
39
27
28$$
29\begin{aligned}
30u(\bm x) &= \sum_{j=1}^n u_j \, \phi_j(\bm x) ,\\
31v(\bm x) &= \sum_{i=1}^n v_i \, \phi_i(\bm x) .
32\end{aligned}
33$$ (eq-nodal-values)
34
40The coefficients $\{u_j\}$ and $\{v_i\}$ are the nodal values of $u$
41and $v$, respectively. Inserting the expressions {eq}`eq-nodal-values`
42into {eq}`eq-general-weak-form`, we obtain the inner-products
35The coefficients $\{u_j\}$ and $\{v_i\}$ are the nodal values of $u$ and $v$, respectively.
36Inserting the expressions {eq}`eq-nodal-values` into {eq}`eq-general-weak-form`, we obtain the inner-products
43
44$$
45\langle v,u \rangle = \bm v^T M \bm u , \qquad \langle v,f\rangle = \bm v^T \bm b \,.
46$$ (eq-inner-prods)
47
37
38$$
39\langle v,u \rangle = \bm v^T M \bm u , \qquad \langle v,f\rangle = \bm v^T \bm b \,.
40$$ (eq-inner-prods)
41
48Here, we have introduced the mass matrix, $M$, and the right-hand side,
49$\bm b$,
42Here, we have introduced the mass matrix, $M$, and the right-hand side, $\bm b$,
50
51$$
52M_{ij} := (\phi_i,\phi_j), \;\; \qquad b_{i} := \langle \phi_i, f \rangle,
53$$
54
55each defined for index sets $i,j \; \in \; \{1,\dots,n\}$.
56
57(laplace-operator)=
58
59## Laplace's Operator
60
43
44$$
45M_{ij} := (\phi_i,\phi_j), \;\; \qquad b_{i} := \langle \phi_i, f \rangle,
46$$
47
48each defined for index sets $i,j \; \in \; \{1,\dots,n\}$.
49
50(laplace-operator)=
51
52## Laplace's Operator
53
61The Laplace's operator used in BP3-BP6 is defined via the following variational
62formulation, i.e., find $u \in V^p$ such that for all $v \in V^p$
54The Laplace's operator used in BP3-BP6 is defined via the following variational formulation, i.e., find $u \in V^p$ such that for all $v \in V^p$
63
64$$
65a(v,u) = \langle v,f \rangle , \,
66$$
67
55
56$$
57a(v,u) = \langle v,f \rangle , \,
58$$
59
68where now $a (v,u)$ expresses the continuous bilinear form defined on
69$V^p$ for sufficiently regular $u$, $v$, and $f$, that is:
60where now $a (v,u)$ expresses the continuous bilinear form defined on $V^p$ for sufficiently regular $u$, $v$, and $f$, that is:
70
71$$
72\begin{aligned} a(v,u) &:= \int_{\Omega}\nabla v \, \cdot \, \nabla u \, dV ,\\ \langle v,f \rangle &:= \int_{\Omega} \, v \, f \, dV . \end{aligned}
73$$
74
61
62$$
63\begin{aligned} a(v,u) &:= \int_{\Omega}\nabla v \, \cdot \, \nabla u \, dV ,\\ \langle v,f \rangle &:= \int_{\Omega} \, v \, f \, dV . \end{aligned}
64$$
65
75After substituting the same formulations provided in {eq}`eq-nodal-values`,
76we obtain
66After substituting the same formulations provided in {eq}`eq-nodal-values`, we obtain
77
78$$
79a(v,u) = \bm v^T K \bm u ,
80$$
81
82in which we have introduced the stiffness (diffusion) matrix, $K$, defined as
83
84$$
85K_{ij} = a(\phi_i,\phi_j),
86$$
87
88for index sets $i,j \; \in \; \{1,\dots,n\}$.
67
68$$
69a(v,u) = \bm v^T K \bm u ,
70$$
71
72in which we have introduced the stiffness (diffusion) matrix, $K$, defined as
73
74$$
75K_{ij} = a(\phi_i,\phi_j),
76$$
77
78for index sets $i,j \; \in \; \{1,\dots,n\}$.