# A priori error analysis for transient problems using Enhanced Velocity approach in the discrete-time setting

## TL;DR

This paper provides theoretical error estimates for transient subsurface flow problems using the Enhanced Velocity Mixed FEM combined with backward Euler and Crank-Nicolson methods, aiding in optimal time and space discretization.

## Contribution

It introduces a priori error analysis in discrete-time for transient problems using Enhanced Velocity Mixed FEM, extending previous adaptive mesh refinement applications.

## Key findings

- Validated backward Euler error estimates through numerical tests
- Error estimates assist in choosing appropriate time step and mesh size
- Enhanced Velocity scheme is effective for heterogeneous porous media

## Abstract

Time discretization along with space discretization is important in the numerical simulation of subsurface flow applications for long run. In this paper, we derive theoretical convergence error estimates in discrete-time setting for transient problems with the Dirichlet boundary condition. Enhanced Velocity Mixed FEM as domain decomposition method is used in the space discretization and the backward Euler method and the Crank-Nicolson method are considered in the discrete-time setting. Enhanced Velocity scheme was used in the adaptive mesh refinement dealing with heterogeneous porous media [1, 2] for single phase flow and transport and demonstrated as mass conservative and efficient method. Numerical tests validating the backward Euler theory are presented. This error estimates are useful in the determining of time step size and the space discretization size. References. [1] Yerlan Amanbek, Gurpreet Singh, Mary F Wheeler, and Hans van Duijn. Adaptive numerical homogenization for upscaling single phase flow and transport. ICES Report,12:17, 2017. [2] Gurpreet Singh, Yerlan Amanbek, and Mary F Wheeler. Adaptive homogenization for upscaling heterogeneous porous medium. In SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, 2017.