# On subgrid multiscale stabilized finite element method for advection-diffusion-reaction equation with variable coefficients

## TL;DR

This paper develops a stabilized finite element method using subgrid scales for solving advection-diffusion-reaction equations with variable coefficients, providing error estimates and a stabilization parameter expression verified through numerical experiments.

## Contribution

Introduces a subgrid multiscale stabilized finite element method with algebraic sub-scale approximation for variable coefficient advection-diffusion-reaction equations, including error analysis and stabilization parameter derivation.

## Key findings

- Error estimates in $L_2$ norms confirm method accuracy.
- Derived stabilization parameter improves numerical stability.
- Numerical experiments validate theoretical results.

## Abstract

In this study a stabilized finite element method for solving advection-diffusion-reaction equation with spatially variable coefficients has been carried out. Here subgrid scale approach along with algebraic approximation to the sub-scales has been chosen as stabilized method among various other methods. Both a priori and a posteriori finite element error estimates in $L_2$ norms have been derived after introducing the stabilized formulation of the variational form. An expression of the stabilization parameter for this 2D problem has also been derived here. At last numerical experiments are presented to verify numerical performance of the stabilized method and check the credibility of the theoretically derived expression of the stabilization parameter.