Decoupling mixed finite elements on hierarchical triangular grids for parabolic problems
This work addresses the efficient solution of parabolic mixed flow problems for computational scientists, but the contribution is incremental as it combines existing techniques.
The paper proposes a numerical method for time-dependent flow problems using decoupled mixed finite elements on hierarchical triangular grids, achieving a cell-centered scheme with a local 10-point stencil and parallel solvability. Numerical experiments demonstrate the algorithm's behavior.
In this paper, we propose a numerical method for the solution of time-dependent flow problems in mixed form. Such problems can be efficiently approximated on hierarchical grids, obtained from an unstructured coarse triangulation by using a regular refinement process inside each of the initial coarse elements. If these elements are considered as subdomains, we can formulate a non-overlapping domain decomposition method based on the lowest-order Raviart-Thomas elements, properly enhanced with Lagrange multipliers on the boundaries of each subdomain (excluding the Dirichlet edges). A suitable choice of mixed finite element spaces and quadrature rules yields a cell-centered scheme for the pressures with a local 10-point stencil. The resulting system of differential-algebraic equations is integrated in time by the Crank-Nicolson method, which is known to be a stiffly accurate scheme. As a result, we obtain independent subdomain linear systems that can be solved in parallel. The behaviour of the algorithm is illustrated on a variety of numerical experiments.