Semi-explicit discretization schemes for weakly coupled elliptic-parabolic problems

Altmann, Robert; Maier, Roland; Unger, Benjamin

doi:10.1090/mcom/3608

Cited by 16 publications

(45 citation statements)

References 36 publications

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“…We consider a 3 × 3 system in order to test the semi-explicit discretization for a nonlinear system, which is motivated by the nonlinear permeability considered in [4]. In the linear setting, we consider finite-dimensional spaces This example satisfies the weak coupling condition proposed in [2] and we refer to this paper for a numerical study. In the linear case, the semi-explicit Euler schemes approximates the true solution well with the same order as the fully implicit scheme.…”

Section: Numerical Toy Examplementioning

confidence: 99%

“…Numerical approximations of (1) usually consider an implicit Euler scheme combined with finite elements, leading to a large (nonlinear) system that has to be solved in each time step. In the linear case, the coupled system can be decoupled with a semi-explicit approach as proposed in [2] (based on observations made in [1]) without decreasing the convergence order in time. To ensure convergence, however, one needs a weak coupling condition in the sense that the continuity constant of d is small compared to the ellipticity constants of a and c. This contribution aims to sound the applicability of semi-explicit discretizations also for the nonlinear model (1) since this would improve the computational efficiency dramatically.…”

Section: Introductionmentioning

confidence: 99%

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A semi‐explicit integration scheme for weakly‐coupled poroelasticity with nonlinear permeability

Altmann

Maier

Unger

2021

Proc Appl Math and Mech

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Within the time integration of linear elliptic-parabolic problems such as poroelasticity, large systems have to be solved in every time step. With a semi-explicit approach, these systems decouple, leading to a remarkable speed-up. This paper indicates that this idea can also be applied to nonlinear problems such as poroelasticity with nonlinear permeability. To see this, we consider a related toy problem.

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Section: Numerical Toy Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A semi‐explicit integration scheme for weakly‐coupled poroelasticity with nonlinear permeability

Altmann

Maier

Unger

2021

Proc Appl Math and Mech

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“…In order to prove convergence of the semi-explicit scheme, we follow the idea first presented in [AMU21a] for the linear case and consider a delay system, which is closely related to the original system (2.4). This is the subject of the following subsection.…”

Section: Time Discretizationmentioning

confidence: 99%

“…Proof of Proposition 3.5. The proof is based on [AMU21a] and follows the ideas of [EM09]. We set η n u := ūn − u n ∈ V and η n p := pn − p n ∈ Q as well as…”

Section: Time Discretizationmentioning

confidence: 99%

“…We refer to this particular configuration as weak coupling, which is satisfied in many applications. The advantage of a semi-explicit discretization for the classical (linear) poroelastic model was studied in [AMU21a]. Therein it was shown that a semi-explicit Euler discretization maintains the first-order convergence and at the same time decouples the system equations.…”

Section: Introductionmentioning

confidence: 99%

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A decoupling and linearizing discretization for weakly coupled poroelasticity with nonlinear permeability

Altmann¹,

Maier²

2021

Preprint

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We analyze a semi-explicit time discretization scheme of first order for poroelasticity with nonlinear permeability provided that the elasticity model and the flow equation are only weakly coupled. The approach leads to a decoupling of the equations and, at the same time, linearizes the nonlinearity without the need of further inner iteration steps. Hence, the computational speed-up is twofold without a loss in the convergence rate. We prove optimal first-order error estimates by considering a related delay system and investigate the method numerically for different examples with various types of nonlinear displacement-permeability relations.

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Higher-order iterative decoupling for poroelasticity

Altmann,

Mujahid,

Unger

2024

Adv Comput Math

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For the iterative decoupling of elliptic–parabolic problems such as poroelasticity, we introduce time discretization schemes up to order five based on the backward differentiation formulae. Its analysis combines techniques known from fixed-point iterations with the convergence analysis of the temporal discretization. As the main result, we show that the convergence depends on the interplay between the time step size and the parameters for the contraction of the iterative scheme. Moreover, this connection is quantified explicitly, which allows for balancing the single error components. Several numerical experiments illustrate and validate the theoretical results, including a three-dimensional example from biomechanics.

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Semi-explicit discretization schemes for weakly coupled elliptic-parabolic problems

Cited by 16 publications

References 36 publications

A semi‐explicit integration scheme for weakly‐coupled poroelasticity with nonlinear permeability

A semi‐explicit integration scheme for weakly‐coupled poroelasticity with nonlinear permeability

A decoupling and linearizing discretization for weakly coupled poroelasticity with nonlinear permeability

Higher-order iterative decoupling for poroelasticity

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