Abstract

We perform a followup computational study of the recently proposed space–time first order system least squares ( FOSLS ) method subject to constraints referred to as CFOSLS where we now combine it with the new capability we have developed, namely, parallel adaptive mesh refinement (AMR) in 4D. The AMR is needed to alleviate the high memory demand in the combined space time domain and also allows general (4D) meshes that better follow the physics in space–time. With an extensive set of computational experiments, performed in parallel, we demonstrate the feasibility of the combined space–time AMR approach in both two space plus time and three space plus time dimensions.

Highlights

  • This study is a continuation of previous work, [1,2,3] for discretizations and solvers of time-dependent problems discretized in combined space–time domain

  • There has been a substantial interest for combined space–time approaches, especially in the solvers community motivated by the parallel-in-time approach for solving time-dependent PDEs, which is viewed as a feasible approach for better utilization of the computational power of the generation parallel computers

  • The adaptive mesh refinement (AMR) allows the use of general meshes in 4D without having the notion of time-stepping, having high resolution only in directions of high variation of the physical quantities modeled by the time dependent PDEs of interest

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Summary

Introduction

This study is a continuation of previous work, [1,2,3] for discretizations and solvers of time-dependent problems discretized in combined space–time domain. The CFOSLS approach though, as noted in [3], poses the challenge to the solvers since the FOSLS functionals are not uniformly elliptic and standard multigrid (with geometric coarse spaces) is not algorithmically scalable our tests do show reasonable performance (in timings). This challenge needs to be addressed further, with possibly exploiting ideas from adaptive AMG that can detect problem anisotropies (cf., e.g., [9] and [10]).

Model time dependent PDEs used in our study
Space–time CFOSLS
Variational CFOSLS formulation
Finite element discretization
Negative norm CFOSLS
Alternative formulation for the transport problem
Linear solvers for CFOSLS
Block diagonal preconditioners
Monolithic geometric multigrid preconditioners
Divergence free solvers
Implementation and experimental setup
Example 1
Example 2
Example 3
Conclusion
Full Text
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