1*bcb2dfaeSJed Brown(bps)= 2*bcb2dfaeSJed Brown 3*bcb2dfaeSJed Brown# CEED Bakeoff Problems 4*bcb2dfaeSJed Brown 5*bcb2dfaeSJed Brown```{include} ./README.md 6*bcb2dfaeSJed Brown:start-after: bps-inclusion-marker 7*bcb2dfaeSJed Brown:end-before: bps-exclusion-marker 8*bcb2dfaeSJed Brown``` 9*bcb2dfaeSJed Brown 10*bcb2dfaeSJed Brown(mass-operator)= 11*bcb2dfaeSJed Brown 12*bcb2dfaeSJed Brown## Mass Operator 13*bcb2dfaeSJed Brown 14*bcb2dfaeSJed BrownThe Mass Operator used in BP1 and BP2 is defined via the $L^2$ projection 15*bcb2dfaeSJed Brownproblem, posed as a weak form on a Hilbert space $V^p \subset H^1$, i.e., 16*bcb2dfaeSJed Brownfind $u \in V^p$ such that for all $v \in V^p$ 17*bcb2dfaeSJed Brown 18*bcb2dfaeSJed Brown$$ 19*bcb2dfaeSJed Brown\langle v,u \rangle = \langle v,f \rangle , 20*bcb2dfaeSJed Brown$$ (eq-general-weak-form) 21*bcb2dfaeSJed Brown 22*bcb2dfaeSJed Brownwhere $\langle v,u\rangle$ and $\langle v,f\rangle$ express the continuous 23*bcb2dfaeSJed Brownbilinear and linear forms, respectively, defined on $V^p$, and, for sufficiently 24*bcb2dfaeSJed Brownregular $u$, $v$, and $f$, we have: 25*bcb2dfaeSJed Brown 26*bcb2dfaeSJed Brown$$ 27*bcb2dfaeSJed Brown\begin{aligned} \langle v,u \rangle &:= \int_{\Omega} \, v \, u \, dV ,\\ \langle v,f \rangle &:= \int_{\Omega} \, v \, f \, dV . \end{aligned} 28*bcb2dfaeSJed Brown$$ 29*bcb2dfaeSJed Brown 30*bcb2dfaeSJed BrownFollowing the standard finite/spectral element approach, we formally 31*bcb2dfaeSJed Brownexpand all functions in terms of basis functions, such as 32*bcb2dfaeSJed Brown 33*bcb2dfaeSJed Brown$$ 34*bcb2dfaeSJed Brown\begin{aligned} 35*bcb2dfaeSJed Brownu(\bm x) &= \sum_{j=1}^n u_j \, \phi_j(\bm x) ,\\ 36*bcb2dfaeSJed Brownv(\bm x) &= \sum_{i=1}^n v_i \, \phi_i(\bm x) . 37*bcb2dfaeSJed Brown\end{aligned} 38*bcb2dfaeSJed Brown$$ (eq-nodal-values) 39*bcb2dfaeSJed Brown 40*bcb2dfaeSJed BrownThe coefficients $\{u_j\}$ and $\{v_i\}$ are the nodal values of $u$ 41*bcb2dfaeSJed Brownand $v$, respectively. Inserting the expressions {math:numref}`eq-nodal-values` 42*bcb2dfaeSJed Browninto {math:numref}`eq-general-weak-form`, we obtain the inner-products 43*bcb2dfaeSJed Brown 44*bcb2dfaeSJed Brown$$ 45*bcb2dfaeSJed Brown\langle v,u \rangle = \bm v^T M \bm u , \qquad \langle v,f\rangle = \bm v^T \bm b \,. 46*bcb2dfaeSJed Brown$$ (eq-inner-prods) 47*bcb2dfaeSJed Brown 48*bcb2dfaeSJed BrownHere, we have introduced the mass matrix, $M$, and the right-hand side, 49*bcb2dfaeSJed Brown$\bm b$, 50*bcb2dfaeSJed Brown 51*bcb2dfaeSJed Brown$$ 52*bcb2dfaeSJed BrownM_{ij} := (\phi_i,\phi_j), \;\; \qquad b_{i} := \langle \phi_i, f \rangle, 53*bcb2dfaeSJed Brown$$ 54*bcb2dfaeSJed Brown 55*bcb2dfaeSJed Browneach defined for index sets $i,j \; \in \; \{1,\dots,n\}$. 56*bcb2dfaeSJed Brown 57*bcb2dfaeSJed Brown(laplace-operator)= 58*bcb2dfaeSJed Brown 59*bcb2dfaeSJed Brown## Laplace's Operator 60*bcb2dfaeSJed Brown 61*bcb2dfaeSJed BrownThe Laplace's operator used in BP3-BP6 is defined via the following variational 62*bcb2dfaeSJed Brownformulation, i.e., find $u \in V^p$ such that for all $v \in V^p$ 63*bcb2dfaeSJed Brown 64*bcb2dfaeSJed Brown$$ 65*bcb2dfaeSJed Browna(v,u) = \langle v,f \rangle , \, 66*bcb2dfaeSJed Brown$$ 67*bcb2dfaeSJed Brown 68*bcb2dfaeSJed Brownwhere now $a (v,u)$ expresses the continuous bilinear form defined on 69*bcb2dfaeSJed Brown$V^p$ for sufficiently regular $u$, $v$, and $f$, that is: 70*bcb2dfaeSJed Brown 71*bcb2dfaeSJed Brown$$ 72*bcb2dfaeSJed Brown\begin{aligned} a(v,u) &:= \int_{\Omega}\nabla v \, \cdot \, \nabla u \, dV ,\\ \langle v,f \rangle &:= \int_{\Omega} \, v \, f \, dV . \end{aligned} 73*bcb2dfaeSJed Brown$$ 74*bcb2dfaeSJed Brown 75*bcb2dfaeSJed BrownAfter substituting the same formulations provided in {math:numref}`eq-nodal-values`, 76*bcb2dfaeSJed Brownwe obtain 77*bcb2dfaeSJed Brown 78*bcb2dfaeSJed Brown$$ 79*bcb2dfaeSJed Browna(v,u) = \bm v^T K \bm u , 80*bcb2dfaeSJed Brown$$ 81*bcb2dfaeSJed Brown 82*bcb2dfaeSJed Brownin which we have introduced the stiffness (diffusion) matrix, $K$, defined as 83*bcb2dfaeSJed Brown 84*bcb2dfaeSJed Brown$$ 85*bcb2dfaeSJed BrownK_{ij} = a(\phi_i,\phi_j), 86*bcb2dfaeSJed Brown$$ 87*bcb2dfaeSJed Brown 88*bcb2dfaeSJed Brownfor index sets $i,j \; \in \; \{1,\dots,n\}$. 89