Structure‐preserving space‐time discretization of a mixed formulation for quasi‐incompressible large strain elasticity in principal stretches

2019 ◽  
Vol 120 (13) ◽  
pp. 1381-1410
Author(s):  
Alexander Janz ◽  
Peter Betsch ◽  
Marlon Franke
Keyword(s):  
Large Strain ◽  
Time Discretization ◽  
Space Time ◽  
Mixed Formulation ◽  
Structure Preserving ◽  
Principal Stretches

scholarly journals Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen–Cahn Equation with Multiplicative Noise

2018 ◽  
Vol 18 (2) ◽  
pp. 297-311 ◽  
Author(s):  
Ananta K. Majee ◽  
Andreas Prohl
Keyword(s):  
Multiplicative Noise ◽  
Time Discretization ◽  
Rates Of Convergence ◽  
Monotonicity Property ◽  
Space Time ◽  
Uniform Bounds ◽  
Structure Preserving ◽  
Cahn Equation ◽  
Weak Monotonicity ◽  
Spatio Temporal

AbstractThe stochastic Allen–Cahn equation with multiplicative noise involves the nonlinear drift operator {{\mathscr{A}}(x)=\Delta x-(|x|^{2}-1)x}. We use the fact that {{\mathscr{A}}(x)=-{\mathcal{J}}^{\prime}(x)} satisfies a weak monotonicity property to deduce uniform bounds in strong norms for solutions of the temporal, as well as of the spatio-temporal discretization of the problem. This weak monotonicity property then allows for the estimate\sup_{1\leq j\leq J}{\mathbb{E}}[\|X_{t_{j}}-Y^{j}\|_{{\mathbb{L}}^{2}}^{2}]% \leq C_{\delta}(k^{1-\delta}+h^{2})for all small {\delta>0}, where X is the strong variational solution of the stochastic Allen–Cahn equation, while {\{Y^{j}:0\leq j\leq J\}} solves a structure preserving finite element based space-time discretization of the problem on a temporal mesh {\{t_{j}:1\leq j\leq J\}} of size {k>0} which covers {[0,T]}.

scholarly journals A Banach space mixed formulation for the unsteady Brinkman–Forchheimer equations

2020 ◽  
Author(s):  
Sergio Caucao ◽  
Ivan Yotov
Keyword(s):  
Banach Space ◽  
Time Discretization ◽  
Monotone Operators ◽  
Rates Of Convergence ◽  
Mixed Formulation ◽  
Model Parameters ◽  
Weak Formulation ◽  
Finite Element Approximations ◽  
Backward Euler ◽  
Discrete Finite Element

Abstract We propose and analyse a mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient and pressure, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well posedness and error analysis for semidiscrete continuous-in-time and fully discrete finite element approximations on simplicial grids with spatial discretization based on the Raviart–Thomas spaces of degree $k$ for the pseudostress tensor and discontinuous piecewise polynomial elements of degree $k$ for the velocity and backward Euler time discretization. We provide several numerical results to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of model parameters.

A stabilized space-time discretization for the primitive equations in oceanography

2004 ◽  
Vol 98 (3) ◽  
pp. 427-475 ◽  
Author(s):  
T. Chacón Rebollo ◽  
D. Rodríguez Gómez
Keyword(s):  
Time Discretization ◽  
Space Time ◽  
Primitive Equations
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