1bcb2dfaeSJed Brown(bps)= 2bcb2dfaeSJed Brown 3bcb2dfaeSJed Brown# CEED Bakeoff Problems 4bcb2dfaeSJed Brown 5bcb2dfaeSJed Brown```{include} ./README.md 6*525f58efSJeremy L Thompson:start-after: <!-- bps-inclusion --> 7*525f58efSJeremy L Thompson:end-before: <!-- bps-exclusion --> 8bcb2dfaeSJed Brown``` 9bcb2dfaeSJed Brown 10bcb2dfaeSJed Brown(mass-operator)= 11bcb2dfaeSJed Brown 12bcb2dfaeSJed Brown## Mass Operator 13bcb2dfaeSJed Brown 1417be3a41SJeremy L ThompsonThe 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 $v \in V^p$ 15bcb2dfaeSJed Brown 16bcb2dfaeSJed Brown$$ 17bcb2dfaeSJed Brown\langle v,u \rangle = \langle v,f \rangle , 18bcb2dfaeSJed Brown$$ (eq-general-weak-form) 19bcb2dfaeSJed Brown 2017be3a41SJeremy L Thompsonwhere $\langle v,u\rangle$ 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: 21bcb2dfaeSJed Brown 22bcb2dfaeSJed Brown$$ 23bcb2dfaeSJed Brown\begin{aligned} \langle v,u \rangle &:= \int_{\Omega} \, v \, u \, dV ,\\ \langle v,f \rangle &:= \int_{\Omega} \, v \, f \, dV . \end{aligned} 24bcb2dfaeSJed Brown$$ 25bcb2dfaeSJed Brown 2617be3a41SJeremy L ThompsonFollowing the standard finite/spectral element approach, we formally expand all functions in terms of basis functions, such as 27bcb2dfaeSJed Brown 28bcb2dfaeSJed Brown$$ 29bcb2dfaeSJed Brown\begin{aligned} 30bcb2dfaeSJed Brownu(\bm x) &= \sum_{j=1}^n u_j \, \phi_j(\bm x) ,\\ 31bcb2dfaeSJed Brownv(\bm x) &= \sum_{i=1}^n v_i \, \phi_i(\bm x) . 32bcb2dfaeSJed Brown\end{aligned} 33bcb2dfaeSJed Brown$$ (eq-nodal-values) 34bcb2dfaeSJed Brown 3517be3a41SJeremy L ThompsonThe coefficients $\{u_j\}$ and $\{v_i\}$ are the nodal values of $u$ and $v$, respectively. 3617be3a41SJeremy L ThompsonInserting the expressions {eq}`eq-nodal-values` into {eq}`eq-general-weak-form`, we obtain the inner-products 37bcb2dfaeSJed Brown 38bcb2dfaeSJed Brown$$ 39bcb2dfaeSJed Brown\langle v,u \rangle = \bm v^T M \bm u , \qquad \langle v,f\rangle = \bm v^T \bm b \,. 40bcb2dfaeSJed Brown$$ (eq-inner-prods) 41bcb2dfaeSJed Brown 4217be3a41SJeremy L ThompsonHere, we have introduced the mass matrix, $M$, and the right-hand side, $\bm b$, 43bcb2dfaeSJed Brown 44bcb2dfaeSJed Brown$$ 45bcb2dfaeSJed BrownM_{ij} := (\phi_i,\phi_j), \;\; \qquad b_{i} := \langle \phi_i, f \rangle, 46bcb2dfaeSJed Brown$$ 47bcb2dfaeSJed Brown 48bcb2dfaeSJed Browneach defined for index sets $i,j \; \in \; \{1,\dots,n\}$. 49bcb2dfaeSJed Brown 50bcb2dfaeSJed Brown(laplace-operator)= 51bcb2dfaeSJed Brown 52bcb2dfaeSJed Brown## Laplace's Operator 53bcb2dfaeSJed Brown 5417be3a41SJeremy L ThompsonThe 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 $v \in V^p$ 55bcb2dfaeSJed Brown 56bcb2dfaeSJed Brown$$ 57bcb2dfaeSJed Browna(v,u) = \langle v,f \rangle , \, 58bcb2dfaeSJed Brown$$ 59bcb2dfaeSJed Brown 6017be3a41SJeremy L Thompsonwhere now $a (v,u)$ expresses the continuous bilinear form defined on $V^p$ for sufficiently regular $u$, $v$, and $f$, that is: 61bcb2dfaeSJed Brown 62bcb2dfaeSJed Brown$$ 63bcb2dfaeSJed Brown\begin{aligned} a(v,u) &:= \int_{\Omega}\nabla v \, \cdot \, \nabla u \, dV ,\\ \langle v,f \rangle &:= \int_{\Omega} \, v \, f \, dV . \end{aligned} 64bcb2dfaeSJed Brown$$ 65bcb2dfaeSJed Brown 6617be3a41SJeremy L ThompsonAfter substituting the same formulations provided in {eq}`eq-nodal-values`, we obtain 67bcb2dfaeSJed Brown 68bcb2dfaeSJed Brown$$ 69bcb2dfaeSJed Browna(v,u) = \bm v^T K \bm u , 70bcb2dfaeSJed Brown$$ 71bcb2dfaeSJed Brown 72bcb2dfaeSJed Brownin which we have introduced the stiffness (diffusion) matrix, $K$, defined as 73bcb2dfaeSJed Brown 74bcb2dfaeSJed Brown$$ 75bcb2dfaeSJed BrownK_{ij} = a(\phi_i,\phi_j), 76bcb2dfaeSJed Brown$$ 77bcb2dfaeSJed Brown 78bcb2dfaeSJed Brownfor index sets $i,j \; \in \; \{1,\dots,n\}$. 79