A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems

By Alexandre Ern, Sébastien Meunier. 2009

60 citations recorded by the source corpus.

We analyze Euler-Galerkin approximations (conforming finite elements in space and im- plicit Euler in time) to coupled PDE systems in which one dependent variable, say u ,i s governed by an elliptic equation and the other, say p, by a parabolic-like equation. The underlying application is the poroelasticity system within the quasi-static assumption. Different polynomial orders are used for the u -a ndp-components to obtain optimally convergent ap rioribounds for all the terms in the error energy norm. Then, a residual-type a posteriori error analysis is performed. Upon extending the ideas of Verfurth for the heat equation (Calcolo 40 (2003) 195-212), an optimally convergent bound is derived for the dominant term in the error energy norm and an equivalence result between residual and error is proven. The error bound can be classically split into time error, space error and data oscillation terms. Moreover, upon extending the elliptic reconstruction technique introduced by Makridakis and Nochetto (SIAM J. Numer. Anal. 41 (2003) 1585-1594), an optimally convergent bound is derived for the remaining terms in the error energy norm. Numerical results are presented to illustrate the theoretical analysis.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/f478d370b268f21029453fdd482fbeb2fa0985a1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Numerical modeling of hydraulic fracture problem in permeable medium using cohesive zone model (2012, methodology): “The choice of P2/P1 interpolation for u−p formulations was shown by [20] to guarantee an optimal convergence of the error estimators in poroelasticity and to reduce the oscillations of the numerical solution when sudden boundary conditions are applied.”
On the optimization of the fixed‐stress splitting for Biot's equations (2018, background): “nuous Galerkin [10, 11] or multiscale methods [12, 13, 14]. Continuous and discontinuous higher-order Galerkin space time elements were proposed in [15]. Adaptive computations were considered e.g. in [16]. A Monte-Carlo approach was proposed in [17]. For a discussion on the stability of the dierent ”