On the accuracy of the mortar finite volume element method with Lagrange multipliers
The authors consider the following elliptic boundary value problem \[ \begin{aligned} -\nabla\cdot(A\nabla u) = f \quad & \quad \text{in }\Omega, \\ u = 0 \quad & \quad \text{on }\partial\Omega, \end{aligned} \] where \(\Omega\subset \mathbb{R}^{2}\) is a bounded convex polygonal domain with boundary \(\partial\Omega\), \(A=(a_{ij}(x))_{2\times 2}\) is a given symmetric and uniformly positive definite matrix in \(\Omega\), \(a_{ij}\in C^{1}(\overline{\Omega})\), \(1\leq i,j\leq 2\), and the source term \(f = f(x)\) is regular enough such that the problem has a unique solution in a certain Sobolev space. The authors investigate the accuracy of the mortar finite volume element (MFVE) method for the problem stated above. They consider the MFVE scheme which follows the traditional mortar approach and use finite volume element approximations only in the subdomains and finite element in the interfaces for Lagrange multipliers. The authors extend the \(H^{1}\)-norm error estimate of \textit{R. Ewing, R. Lazarov, T. Lin}, and \textit{Y. Lin} [East-West J. Numer. Math. 8, No. 2, 93--110 (2000; Zbl 0963.65112)] and prove the general \(L^{2}\)-norm error estimate to show how the regularity of the source term \(f\) can affect the convergence rate of the MFVE solution. Finally, the uniform convergence of the solution for the MFVE method under minimal regularity assumption is presented.
mortar finite volume element (MFVE) method
scientific article; zbMATH DE number 5128324
scientific article