The Hermite Finite Volume Method with Global Conservation Law

We construct a high-order (cubic) Hermite finite volume method (FVM-2L) with a two-layered dual strategy on triangular meshes, which possesses the conservation laws in both flux form and equation form. In particular, for problems with Dirichlet boundary conditions, the FVM-2L scheme preserves conservation laws on all triangles, whereas conservation properties may be lost on boundary dual elements by existing vertex-centered finite volume schemes. Theoretically, this is the first L^2 result for the Hermite finite volume method on triangular meshes. Furthermore, the regularity requirement for the L^2 theory of the FVM-2L scheme is reduced to u∈ H^k+1 (i.e. u∈ H^4 ). While, as a comparison, all existing L^2 results for high-order ( k≥ 2 ) finite volume schemes require u∈ H^k+2 in the analysis. Finally, the conservation and convergence properties of the FVM-2L scheme are verified numerically for a selection of elliptic, linear elastic, Stokes, and heat conduction problems.