Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

Abstract

We consider in this paper the time-dependent two-phase Stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward Euler temporal discretizations. Regularization of the enthalpy-temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered. Our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual, as well as on the L 2 (L 2 ) error of the temperature and the L 2 (H −1 ) error of the enthalpy. Moreover, they allow to distinguish the space, time, regularization, and linearization error components. An adaptive algorithm is proposed, which ensures computational savings through the online choice of a sufficient regularization parameter, a stopping criterion for the linearization iterations, local space mesh refinement, time step adjustment, and equilibration of the spatial and temporal errors. We also prove the efficiency of our estimate. Our analysis is quite general and is not focused on a specific choice of the space discretization and of the linearization. As an example, we apply it to the vertex-centered finite volume (finite element with mass lumping and quadrature) and Newton methods. Numerical results illustrate the effectiveness of our estimates and the performance of the adaptive algorithm.MSC: 65N08, 65N15, 65N50, 80A22 IntroductionThe two-phase Stefan problem models a phase change process which is governed by the Fourier law, c.f. Friedman [22]. The two phases, typically solid and liquid, are separated by a moving interface, whose motion is governed by the so-called Stefan condition., be an open bounded polygonal or polyhedral domain, not necessarily convex, and let T > 0. The mathematical statement of the problem is as follows: given an initial enthalpy u 0 and a source function f , find the enthalpy u such that(1.1c) * This work was supported by the ERT project "Enhanced oil recovery and geological sequestration of CO 2 : mesh adaptivity, a posteriori error control, and other advanced techniques" (LJLL/IFPEN) † daniele.di-pietro@univ-montp2.fr ‡

Reservoir Simulator Runtime Enhancement Based ona PosterioriError Estimation Techniques

“…with R n;k;i p;M defined by (7). To complete both (11b) and (11c), we set respectively H n;k;i lin;p;h Á n X ¼ 0 and H n;k;i alg;p;h Á n X ¼ 0 on oX.…”

“…First rigorous approachs can be found in [2][3][4] suitable for unsteady nonlinear models, and [5][6][7] for degenerated problems. Recently, abstract frameworks for a posteriori estimates appeared, for two-phase flow [8,9], multiphase compositional model [10], and the thermal multiphase compositional global model [11].…”

A Review of Recent Advances in Discretization Methods,a PosterioriError Analysis, and Adaptive Algorithms for Numerical Modeling in Geosciences

“…Moreover, this theory has been unified such that it can be presented independently of the particular numerical discretization. We use such a spirit here, while following the recent contributions [27,[83][84][85][86][87][88][89][90][91][92][93]. The basic idea of this approach can be traced back at least to the Prager and Synge equality [94] and has been used in a posteriori error estimation from the 70s; we refer for a general orientation to the monographs [23,[95][96][97][98] and for milestone contributions [24][25][26][99][100][101][102][103][104][105].…”

“…Refining and derefining the mesh adaptively while following the front (and choosing adaptively the time step size) is likely to still increase the computational attractiveness of our approach. One example of a simpler model problem with similar numerical difficulties for which an entirely adaptive algorithm has already been successfully put in place is discussed in [84].…”

An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow

“…The remaining degrees of freedom can be specified in various ways, as discussed in [54,23,25,21], typically by the solution of local (Dirichlet-Neumann) problems by the mixed finite element method or by direct prescription. So, equations (4.5a)-(4.5b) do not specify u n n,h and u n t,h completely.…”