Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

TL;DR

This paper develops an optimized Schwarz waveform relaxation method with Ventcel transmission conditions for linear advection-diffusion equations, utilizing a mixed hybrid finite element approach and demonstrating improved convergence and flexibility with nonconforming time grids.

Contribution

It introduces a novel combination of Ventcel conditions with a mixed hybrid finite element method for efficient domain decomposition in transport problems.

Findings

Validated accuracy with discontinuous coefficients and various Peclét numbers.

Demonstrated improved convergence over Robin conditions.

Enabled different time steps in subdomains for flexibility.

Abstract

This paper is concerned with the optimized Schwarz waveform relaxation method and Ventcel transmission conditions for the linear advection-diffusion equation. A mixed formulation is considered in which the flux variable represents both diffusive and advective flux, and Lagrange multipliers are introduced on the interfaces between nonoverlapping subdomains to handle tangential derivatives in the Ventcel conditions. A space-time interface problem is formulated and is solved iteratively. Each iteration involves the solution of time-dependent problems with Ventcel boundary conditions in the subdomains. The subdomain problems are discretized in space by a mixed hybrid finite element method based on the lowest-order Raviart-Thomas space and in time by the backward Euler method. The proposed algorithm is fully implicit and enables different time steps in the subdomains. Numerical results with discontinuous coefficients and various Pecl\'et numbers validate the accuracy of the method with nonconforming time grids and confirm the improved convergence properties of Ventcel conditions over Robin conditions.