# Stability of Galerkin discretizations of a mixed space-time variational formulation of parabolic evolution equations

## TL;DR

This paper investigates the stability of Galerkin discretizations for a new mixed space-time variational formulation of parabolic PDEs, demonstrating uniform stability for suitable finite element spaces and comparing with existing methods.

## Contribution

It introduces a new well-posed mixed space-time variational formulation and proves uniform stability of its Galerkin discretizations, advancing numerical analysis of parabolic PDEs.

## Key findings

- Galerkin operators are uniformly stable for suitable trial spaces
- Comparison with existing space-time discretization methods
- Provides theoretical foundation for stable numerical schemes

## Abstract

We analyze Galerkin discretizations of a new well-posed mixed space-time variational formulation of parabolic PDEs. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The method is compared to two related space-time discretization methods introduced in [IMA J. Numer. Anal., 33(1) (2013), pp. 242-260] by R. Andreev and in [Comput. Methods Appl. Math., 15(4) (2015), pp. 551-566] by O. Steinbach.