# Decoupling mixed finite elements on hierarchical triangular grids for parabolic problems

## TL;DR

This paper introduces a parallelizable mixed finite element method on hierarchical triangular grids for parabolic flow problems, combining domain decomposition, Raviart-Thomas elements, and Crank-Nicolson time integration.

## Contribution

It presents a novel non-overlapping domain decomposition approach with cell-centered pressures and parallelizable subdomain systems for efficient parabolic problem solving.

## Key findings

- Efficient parallel solution of flow problems.
- Cell-centered pressure scheme with a 10-point stencil.
- Stable and accurate time integration with Crank-Nicolson.

## Abstract

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.