A discontinuous in time Streamline Diffusion Virtual Element Method for Darcy-transport problem

TL;DR

This paper introduces a novel discontinuous in time Streamline Diffusion Virtual Element Method for simulating Darcy-transport problems involving reactive species, providing theoretical error estimates and supporting numerical experiments.

Contribution

It develops a new discontinuous in time virtual element method combined with streamline diffusion for Darcy-transport, including error analysis and high-order accuracy validation.

Findings

The method achieves arbitrary-order accuracy in space and time.

Theoretical error estimates are validated by numerical experiments.

The approach effectively models reactive transport phenomena.

Abstract

We present a first numerical study of transport phenomena involving chemically reactive species, modeled by advection-diffusion-reaction systems with flow fields governed by Darcy's law. Among the various discretisation approaches, we consider the Streamline Diffusion method. Both the velocity field and the species concentrations are computed using the Virtual Element Method using a Discontinuous Galerkin scheme for time. An abstract error estimate has been derived using a special technique that utilizes Gauss-Radau interpolation in conjunction with numerical integration. These theoretical findings are supported by numerical experiments with arbitrary-order accuracy in both space and time.