scholarly journals Structure‐preserving space‐time discretization of large‐strain thermo‐viscoelasticity in the framework of GENERIC

2021 ◽  
Author(s):  
Mark Schiebl ◽  
Peter Betsch
Keyword(s):  
Large Strain ◽  
Time Discretization ◽  
Space Time ◽  
Structure Preserving

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]}.

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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