Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems

TL;DR

The paper develops a posteriori error estimators for fully discrete linear parabolic problems on space-time domains with time-varying finite element spaces, using an elliptic reconstruction to separate spatial and temporal errors. By formulating a main parabolic error equation for the time-dependent part and exploiting elliptic residuals and jumps, the authors obtain optimal-order bounds in $L_\infty(0,T;L_2(\Omega))$, $L_\infty(0,T;H^1_0(\Omega))$, and $H^1(0,T;L_2(\Omega))$, without requiring restrictive mesh-change conditions. The estimators combine residual-based spatial indicators with time- and data-approximation terms, including mesh-change effects, and are validated by numerical experiments that show the predicted convergence rates. This work provides a practical and flexible framework for adaptive space-time discretization of parabolic problems, leveraging elliptic theory to inform parabolic error control across multiple norms.

Abstract

We derive aposteriori error estimates for fully discrete approximations to solutions of linear parabolic equations on the space-time domain. The space discretization uses finite element spaces, that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto (2003). We derive novel optimal order aposteriori error estimates for the maximum-in-time and mean-square-in-space norm and the mean-square in space-time of the time-derivative norm.

Figures (4)

• Figure 1: Numerical results for problem with exact solution (\ref{['eqn:benchmark']} slow) with $\mathbb P\xspace^{1}$ elements and $\tau\approx h^2$. The abscissa represents time which ranges in $[0,1]$. In the top-most row we plot the various estimators and in the second row we show the corresonding EOCs. Notice that $\max\varepsilon_{\infty,n}$ has EOC $2$ whereas $\left(\tau\sum\varepsilon_{2,n}^2\right)^{1/2}$ has EOC $1$. These are the leading terms in the total estimators (the 3 and 4 plots in the 3rd row) and match in EOC, respectively, the $\operatorname L\xspace_{\infty}(\operatorname L\xspace_{2})$ error and of the $\operatorname L\xspace_{2}(sobh1)$ error, as shown in the first 2 plots of the 3rd row and the 4th row. Thus \ref{['eqn:lil2.estimate']} and \ref{['eqn:l2h1.estimate']} are seen to be sharp and optimal. The last two plots in the 4th row are the effectivity indexes for each norm.
• Figure 2: Simulation with $\mathbb P\xspace^{1}$ elements and $\tau\approx h$. Dominant time discretization error is created by taking the problem with fast time-oscillating exact solution (\ref{['eqn:benchmark']} fast). The abscissa represents time which ranges in $[0,1]$. In the top-most row we plot the various estimators and in the second row we show the corresonding EOCs. Notice that $\tau\sum\theta_{\infty,n}$ has EOC $1$, as opposed to $2$ in the previous example. This is now a leading term in both total estimators (plots 3 and 4 in the 3rd row) which have both EOC $1$. This EOC matches that of the $\operatorname L\xspace_{\infty}(\operatorname L\xspace_{2})$ error and of the $\operatorname L\xspace_{2}(sobh1)$ error, as shown in plots 1 and 2 of the 3rd row and the 4th row. Thus \ref{['eqn:lil2.estimate']} and \ref{['eqn:l2h1.estimate']} are both sharp, but only the second one is optimal due to the coupling. The last two plots in the 4th row are the effectivity indexes for each norm.
• Figure 3: Numerical results for problem with exact solution (\ref{['eqn:benchmark']} slow) with $\mathbb P\xspace^{1}$ elements and $\tau\approx h^3$. The abscissa represents time which ranges in $[0,1]$. In the top-most row we plot the various estimators and in the second row we show the corresonding EOCs. Notice that $\max\varepsilon_{\infty,n}$ has EOC $3$ whereas $\left(\tau\sum\varepsilon_{2,n}^2\right)^{1/2}$ has EOC $2$. These are the leading terms in the total estimators (the 3 and 4 plots in the 3rd row) and match in EOC, respectively, the $\operatorname L\xspace_{\infty}(\operatorname L\xspace_{2})$ error and of the $\operatorname L\xspace_{2}(sobh1)$ error, as shown in plots 1 and 2 of the 3rd and 4th rows. Here the estimates \ref{['eqn:lil2.estimate']} and \ref{['eqn:l2h1.estimate']} are both sharp and optimal. The last two plots in the 4th row are the effectivity indexes for each norm.
• Figure 4: Simulation with $\mathbb P\xspace^{2}$ elements and coupling $\tau\approx h^2$. The time discretization error is dominant since exact is solution (\ref{['eqn:benchmark']} fast). The abscissa represents time which ranges in $[0,1]$. In the top-most row we plot the various estimators and in the second row we show the corresonding EOCs. Notice that $\tau\sum\theta_{\infty,n}$ has EOC $2$, as opposed to $3$ in the previous example, because of the different coupling of the mesh and step sizes. The time estimator is now a leading term in both total estimators (plots 3 and 4 in the 3rd row) which have both EOC $2$. This EOC matches that of the $\operatorname L\xspace_{\infty}(\operatorname L\xspace_{2})$ error and of the $\operatorname L\xspace_{2}(sobh1)$ error, as shown in plots 1 and 2 of the 3rd row and the 4th row. Thus \ref{['eqn:lil2.estimate']} and \ref{['eqn:l2h1.estimate']} are both sharp, but only the second one is optimal. The last two plots in the 4th row are the effectivity indexes for each norm.

Theorems & Definitions (19)

• Definition 1: fully discrete scheme
• Lemma 1: main parabolic error equation
• Definition 2: representation of the elliptic operator, discrete elliptic operator, projections
• Definition 3: discrete elliptic operator
• Definition 4: elliptic reconstruction
• Lemma 2: elliptic reconstruction error estimates
• Definition 5: discrete time extensions and derivatives
• Remark 1: pointwise form
• Remark 2: How to compute $A^{n}U^n$?
• Definition 6: residuals