Loading...
A posteriori error computation in finite element method for initial value problems
Abboud, Joseph
Abboud, Joseph
Citations
Altmetric:
Abstract
In space-time coupled and in space-time decoupled finite element methods of obtaining solutions of initial value problems, determination of the errors (a-posteriori computations) in the computed solution during the evolution is essential: 1) to ensure that the computed solution is meaningful, 2) if the errors are higher than the accepted threshold, then to be able to design an adaptive process based on these errors to improve the accuracy of the solution.
In this paper we present "a-posteriori" error computation methodologies, computational infrastrature and model problem studies for the finite element solutions of initial value problems (IVPs) obtained using space-time coupled (STC) and space-time decoupled (STDC) finite element methods. In initial value problems, the physics require simultaneous dependence of solutions in space and time. Solutions of initial value problems are evolutions i.e. the solution changes at spatial locations as time elapses. The most efficient way to calculate such solution is to consider increments of time Δt starting with initial condition at time t0 or zero and compute evolution for t0 ≤t ≤Δt. Upon obtaining a converged solution for this increment of time, we can then march to t0+Δt ≤ t ≤ 2Δt and so on.
In space-time decoupled methods, we consider a spatial discretization over the integral form of the initial value problem in space over the discretization. The local approximations consist of approximation dependent on spatial coordinates and the degrees of freedom dependent on time. Using the local approximations integrals over elements are evaluated in space, thereby eliminating spatial coordinates and leaving only time variations of the degrees of freedom. Upon assembly, we obtain a system of ordinary differential equations in time that need to be integrated to obtain solution for dofs and their time derivatives. Thus, in this approach, called space-time decoupled methods, treatment in space and time is not simultaneous as required by the initial value problem.
In both approaches, the solution is obtained for an increment of time. Thus, it is beneficial to consider "a-posteriori" error computations for each increment of time for space-time coupled as well as space-time decoupled finite element processes. In this paper we present details of the mathematical and computational infrastructure for a-posteriori error computations for space-time coupled as well as space-time decoupled methodologies for the solutions of initial value problems. This is followed by model problem studies to illustrate various aspects of a-posteriori error computation methodologies for the solutions of initial value problems obtained from space-time coupled and space-time decoupled finite element methods. Summary, the work presented in this paper and some conclusion drawn from it are presented in the last section of the paper.
Description
Date
2025-05-31
Journal Title
Journal ISSN
Volume Title
Publisher
University of Kansas