Lines Matching +full:- +full:v

6 :start-after: <!-- bps-inclusion -->
7 :end-before: <!-- bps-exclusion -->
10 (mass-operator)=
14 …, posed as a weak form on a Hilbert space $V^p \subset H^1$, i.e., find $u \in V^p$ such that for …
17 \langle v,u \rangle = \langle v,f \rangle ,
18 $$ (eq-general-weak-form)
20 …v,u\rangle$ and $\langle v,f\rangle$ express the continuous bilinear and linear forms, respectivel…
23 \begin{aligned} \langle v,u \rangle &:= \int_{\Omega} \, v \, u \, dV ,\\ \langle v,f \rangle &:= \…
31 v(\bm x) &= \sum_{i=1}^n v_i \, \phi_i(\bm x) .
33 $$ (eq-nodal-values)
35 The coefficients $\{u_j\}$ and $\{v_i\}$ are the nodal values of $u$ and $v$, respectively.
36 Inserting the expressions {eq}`eq-nodal-values` into {eq}`eq-general-weak-form`, we obtain the inne…
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)
42 Here, we have introduced the mass matrix, $M$, and the right-hand side, $\bm b$,
50 (laplace-operator)=
54 …s operator used in BP3-BP6 is defined via the following variational formulation, i.e., find $u \in…
57 a(v,u) = \langle v,f \rangle , \,
60 where now $a (v,u)$ expresses the continuous bilinear form defined on $V^p$ for sufficiently regula…
63 \begin{aligned} a(v,u) &:= \int_{\Omega}\nabla v \, \cdot \, \nabla u \, dV ,\\ \langle v,f \rangle…
66 After substituting the same formulations provided in {eq}`eq-nodal-values`, we obtain
69 a(v,u) = \bm v^T K \bm u ,