Space-time finite element analysis of the advection-diffusion equation using Galerkin/least-square stabilization

TL;DR

This work presents a full space-time numerical solution of the advection-diffusion equation using a continuous Galerkin finite element method and derives a priori error estimates and illustrates spatio-temporal convergence with several numerical examples.

Abstract

We present a full space-time numerical solution of the advection-diffusion equation using a continuous Galerkin finite element method on conforming meshes. The Galerkin/least-square method is employed to ensure stability of the discrete variational problem. In the full space-time formulation, time is considered another dimension, and the time derivative is interpreted as an additional advection term of the field variable. We derive a priori error estimates and illustrate spatio-temporal convergence with several numerical examples. We also derive a posteriori error estimates, which coupled with adaptive space-time mesh refinement provide efficient and accurate solutions. The accuracy of the space-time solutions is illustrated against analytical solutions as well as against numerical solutions using a conventional time-marching algorithm.

Figures (32)

• Figure 1: Schematic depiction of the space-time domain $U = \Omega \times I_T$, where $\Omega \subset \mathbb{R}^{d}$, and $I_T = [0,T] \subset \mathbb{R}^{+}.$
• Figure 2: ${\mathbold{a}} = (0,0),\ \nu=10^{-2}$
• Figure 3: ${\mathbold{a}} = 2\pi (-y+1/2,\ x-1/2),\ \nu=10^{-2}$
• Figure 4: ${\mathbold{a}} = (0,0),\ \nu=10^{-6}$
• Figure 5: ${\mathbold{a}} = 2\pi (-y+1/2,\ x-1/2),\ \nu=10^{-6}$

Theorems & Definitions (27)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Lemma 2.1: Boundedness
• proof
• Lemma 2.2: Coercivity
• proof
• Corollary 2.2.1
• proof