Coercivity, monotonicity, conservation, and affine exactness of the mimetic schemes

For an affine $\mathrm{\ell }(x)=b\cdot x+c$, the cell and face means are ${\mathrm{\ell }}_K=\mathrm{\ell }(x_K)$ and ${\mathrm{\ell }}_F=\mathrm{\ell }(x_F)$. The first face-moment identity annihilates the constant term in $G_K\mathrm{\ell }$; the transpose of the second identity is ${\sum }_F\mathrm{\mid }F\mathrm{\mid }{n}_{K,F}\otimes x_F=\mathrm{\mid }K\mathrm{\mid }{I}_d$, so $G_K\mathrm{\ell }=b$. Hence ${\rho }_{K,F}(\mathrm{\ell })={\mathrm{\ell }}_F-{\mathrm{\ell }}_K-b\cdot (x_F-x_K)=0$.

For arbitrary $v$, [A1] gives $\mathrm{\mid }K\mathrm{\mid }(A_K{G}_{K}v)\cdot G_{K}v\ge {\lambda }_{\min }(A_K)\mathrm{\mid }K\mathrm{\mid }\mathrm{\mid }{G}_{K}v{\mathrm{\mid }}^2$. The stabilization coefficient is ${\lambda }_{\min }(A_K)>0$, so summing its squared residual terms gives ${\mathcal{A}}_K(v,v)\ge {\lambda }_{\min }(A_K)\mathrm{\parallel }v{\mathrm{\parallel }}_{K,\mathrm{m}\mathrm{i}\mathrm{m}}^2$. The displayed seminorm is the Euclidean norm of a linear gradient-residual map with positive weights. It vanishes exactly when $G_{K}v=0$ and every residual is zero, hence exactly when all local values equal $v_K$; conversely the first face-moment identity makes both gradient and residual vanish on constants.

For the nonlinear flux, the cell average preserves monotonicity because $(b_K(\xi )-b_K(\eta ))\cdot (\xi -\eta )=\frac{1}{\mathrm{\mid }K\mathrm{\mid }}{\int }_K(a(x,\xi )-a(x,\eta ))\cdot (\xi -\eta )dx\ge 0.$ The pointwise bounds in [A2] also give $b_K(\xi )\cdot \xi \ge \alpha \mathrm{\mid }\xi {\mathrm{\mid }}^p-\mathrm{\mid }K{\mathrm{\mid }}^{-1}{\int }_{K}h$ and $\mathrm{\mid }{b}_K(\xi )\mathrm{\mid }\le C(\mathrm{\mid }K{\mathrm{\mid }}^{-1}{\int }_{K}g+\mathrm{\mid }\xi {\mathrm{\mid }}^{p-1})$. Thus the averaged flux retains monotonicity, coercivity, and $p$-growth.

The global form is symmetric because each $A_K$ is symmetric and the residual term is a symmetric product. If its energy at $v$ is zero, step 1.2 forces $G_{K}v=0$ and every ${\rho }_{K,F}(v)=0$. The latter then gives $v_F=v_K$ on every face. Shared face values make all adjacent cell values equal, and a boundary face has prescribed value zero; boundary anchoring propagates this zero to every cell. Thus the homogeneous matrix has trivial kernel. It is square and finite-dimensional, so it is invertible and positive definite; every cell-source vector has a unique solution.

For an interior face $F=K\mathrm{\mid }L$, the global equation tested by its shared face basis function gives ${\mathcal{A}}_K(u,{\varphi }_{K,F})+{\mathcal{A}}_L(u,{\varphi }_{L,F})=0$, hence $Q_{K,F}+Q_{L,F}=0$. For a cell basis ${\varphi }_K$, the equation gives ${\mathcal{A}}_K(u,{\varphi }_K)={\int }_{K}f$. The local constant vector ${\mathbf{1}}_K={\varphi }_K+{\sum }_{F\subset \mathrm{\partial }K}{\varphi }_{K,F}$ has zero reconstructed gradient by the first face-moment identity and zero residual; therefore ${\mathcal{A}}_K(u,{\mathbf{1}}_K)=0$. Expanding this identity and using the cell equation yields ${\sum }_{F\subset \mathrm{\partial }K}{Q}_{K,F}={\int }_{K}f$.

If $u$ is the face and cell averages of an affine $\mathrm{\ell }$, step 1.1 gives $G_{K}u=b$ and all residuals of $u$ vanish. For the local face basis, $G_K{\varphi }_{K,F}=\mathrm{\mid }F\mathrm{\mid }{n}_{K,F}\mathrm{/}\mathrm{\mid }K\mathrm{\mid }$. Thus ${\mathcal{A}}_K(u,{\varphi }_{K,F})=\mathrm{\mid }F\mathrm{\mid }(A_{K}b)\cdot n_{K,F}$, and the definition $Q_{K,F}=-{\mathcal{A}}_K(u,{\varphi }_{K,F})$ gives the exact outward diffusive flux.

The scalar map ${\psi }_p(s)=\mathrm{\mid }s{\mathrm{\mid }}^{p-2}s$ is increasing: it is increasing on each half-line by monotonicity of $r\mapsto r^{p-1}$ for $r\ge 0$, and its values are nonpositive on $s\le 0$ and nonnegative on $s\ge 0$. Hence $({\psi }_p(r)-{\psi }_p(s))(r-s)\ge 0$. For any two states $u,v$, the difference ${\mathcal{A}}_{h,K}(u;u-v)-{\mathcal{A}}_{h,K}(v;u-v)$ is the sum of $\mathrm{\mid }K\mathrm{\mid }(b_K(G_{K}u)-b_K(G_{K}v))\cdot (G_{K}u-G_{K}v)$ and the nonnegative residual terms ${\alpha }_{0,K}\mathrm{\mid }F\mathrm{\mid }{d}_{K,F}^{1-p}({\psi }_p({\rho }_{K,F}(u))-{\psi }_p({\rho }_{K,F}(v)))({\rho }_{K,F}(u)-{\rho }_{K,F}(v))$. Step 1.3 and scalar monotonicity make every term nonnegative. Testing with $v=u$ and using the coercivity bound in [A2] yields ${\mathcal{A}}_{h,K}(u;u)\ge \mathrm{\mid }K\mathrm{\mid }(\alpha \mathrm{\mid }{G}_{K}u{\mathrm{\mid }}^p-\mathrm{\mid }K{\mathrm{\mid }}^{-1}{\int }_{K}h)+{\alpha }_{0,K}{\sum }_F\frac{\mathrm{\mid }F\mathrm{\mid }}{d_{K,F}^{p-1}}\mathrm{\mid }{\rho }_{K,F}(u){\mathrm{\mid }}^p\mathrm{.}$ For a fixed state, the local constant test vector has zero gradient and residual, so the spatial form annihilates it. A shared interior-face test in an implicit step makes $Q_{K,F}^{\mathrm{n}\mathrm{l}}+Q_{L,F}^{\mathrm{n}\mathrm{l}}=0$. Testing the step equation by the cell basis gives ${\mathcal{A}}_{h,K}(u^n;{\varphi }_K)=\mathrm{\mid }K\mathrm{\mid }{f}_K^n-\mathrm{\mid }K\mathrm{\mid }(u_K^n-u_K^{n-1})\mathrm{/}{\tau }_n$. Expanding the zero constant-test identity gives ${\sum }_F{Q}_{K,F}^{\mathrm{n}\mathrm{l}}=\mathrm{\mid }K\mathrm{\mid }{f}_K^n-\mathrm{\mid }K\mathrm{\mid }(u_K^n-u_K^{n-1})\mathrm{/}{\tau }_n$, the local balance with accumulation.