# A posteriori error analysis for a space-time parallel discretization of parabolic partial differential equations

@article{Chaudhry2021APE, title={A posteriori error analysis for a space-time parallel discretization of parabolic partial differential equations}, author={Jehanzeb Hameed Chaudhry and Donald J. Estep and Simon Tavener}, journal={ArXiv}, year={2021}, volume={abs/2111.00606} }

We construct a space-time parallel method for solving parabolic partial differential equations by coupling the Parareal algorithm in time with overlapping domain decomposition in space. Reformulating the original Parareal algorithm as a variational method and implementing a finite element discretization in space enables an adjoint-based a posteriori error analysis to be performed. Through an appropriate choice of adjoint problems and residuals the error analysis distinguishes between errors…

## Figures and Tables from this paper

## One Citation

### Error estimation for the time to a threshold value in evolutionary partial differential equations

- Computer Science, Mathematics
- 2021

We develop an a posteriori error analysis for a numerical estimate of the time at which a functional of the solution to a partial diﬀerential equation (PDE) ﬁrst achieves a threshold value on a given…

## References

SHOWING 1-10 OF 43 REFERENCES

### A posteriori error analysis for Schwarz overlapping domain decomposition methods

- Computer ScienceBIT Numerical Mathematics
- 2021

An adjoint-based a posteriori error analysis for overlapping multiplicative Schwarz and for overlapping additive Schwarz domain decomposition methods is developed.

### A Posteriori Error Analysis of Two-Stage Computation Methods with Application to Efficient Discretization and the Parareal Algorithm

- Computer ScienceSIAM J. Numer. Anal.
- 2016

The analysis accommodates various variations in the two stage computation and in formulation of the adjoint problems and is applied to compute "dual-weighted" a posteriori error estimates, develop novel algorithms for efficient solution that take into account cancellation of error, and to the Parareal Algorithm.

### A posteriori Error Estimation for the Spectral Deferred Correction Method

- MathematicsJ. Comput. Appl. Math.
- 2021

### Applications of time parallelization

- Computer Science
- 2020

This review article serves to summarize the many advances in time-parallel computations since the excellent review article by Gander, “50 years of Time Parallel Integration” (Gander, in: 50 years of…

### A posteriori analysis of an IMEX entropy-viscosity formulation for hyperbolic conservation laws with dissipation

- MathematicsApplied Numerical Mathematics
- 2019

### An optimal control approach to a posteriori error estimation in finite element methods

- Computer ScienceActa Numerica
- 2001

The ‘dual-weighted-residual method’ is introduced initially within an abstract functional analytic setting, and is then developed in detail for several model situations featuring the characteristic properties of elliptic, parabolic and hyperbolic problems.

### Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality

- MathematicsActa Numerica
- 2002

We give an overview of recent developments concerning the use of adjoint methods in two areas: the a posteriori error analysis of finite element methods for the numerical solution of partial…

### Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations

- Computer Science
- 1996

### Schwarz Alternating Method

- Computer Science
- 1998

A discrete technique of the Schwarz alternating method is presented, to combine the Ritz-Galerkin and finite element methods, well suited for solving singularity problems in parallel.

### Domain-Based Parallelism and Problem Decomposition Methods in Computational Science and Engineering

- Computer Science
- 1995

This refereed volume arose from the editors' recognition that physical scientists, engineers, and applied mathematicians are developing, in parallel, solutions to problems of parallelization and focuses on transferable algorithmic techniques, rather than the scientific results themselves.