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 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 $v \in V^p$ 15 16$$ 17\langle v,u \rangle = \langle v,f \rangle , 18$$ (eq-general-weak-form) 19 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: 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 26Following the standard finite/spectral element approach, we formally expand all functions in terms of basis functions, such as 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 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 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 42Here, we have introduced the mass matrix, $M$, and the right-hand side, $\bm b$, 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 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 $v \in V^p$ 55 56$$ 57a(v,u) = \langle v,f \rangle , \, 58$$ 59 60where now $a (v,u)$ expresses the continuous bilinear form defined on $V^p$ for sufficiently regular $u$, $v$, and $f$, that is: 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 66After substituting the same formulations provided in {eq}`eq-nodal-values`, we obtain 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 $i,j \; \in \; \{1,\dots,n\}$. 79