Erratum to: Weak Formulation of Evolution Problems

2015 ◽  
pp. 699-700
Author(s):  
Sandro Salsa
Keyword(s):  
Weak Formulation ◽  
Evolution Problems

Stability of Petrov–Galerkin Discretizations: Application to the Space-Time Weak Formulation for Parabolic Evolution Problems

2014 ◽  
Vol 14 (2) ◽  
pp. 231-255 ◽  
Author(s):  
Christian Mollet
Keyword(s):  
A Priori ◽  
Uniform Boundedness ◽  
Basis Property ◽  
Weak Form ◽  
Space Time ◽  
Weak Formulation ◽  
Riesz Basis Property ◽  
Evolution Problems ◽  
The Stability ◽  
Discrete Spaces

Abstract. This paper is concerned with the stability of Petrov–Galerkin discretizations with application to parabolic evolution problems in space-time weak form. We will prove that the discrete inf-sup condition for an a priori fixed Petrov–Galerkin discretization is satisfied uniformly under standard approximation and smoothness conditions without any further coupling between the discrete trial and test spaces for sufficiently regular operators. It turns out that one needs to choose different discretization levels for the trial and test spaces in order to obtain a positive lower bound for the discrete inf-sup condition which is independent of the discretization levels. In particular, we state the required number of extra layers in order to guarantee uniform boundedness of the discrete inf-sup constants explicitly. This general result will be applied to the space-time weak formulation of parabolic evolution problems as an important model example. In this regard, we consider suitable hierarchical families of discrete spaces. The results apply, e.g., for finite element discretizations as well as for wavelet discretizations. Due to the Riesz basis property, wavelet discretizations allow for optimal preconditioning independently of the grid spacing. Moreover, our predictions on the stability, especially in view of the dependence on the refinement levels w.r.t. the test and trial spaces, are underlined by numerical results. Furthermore, it can be observed that choosing the same discretization levels would, indeed, lead to stability problems.

Weak formulation of evolution problems

2013 ◽  
pp. 359-386
Author(s):  
Sandro Salsa ◽  
Federico M. G. Vegni ◽  
Anna Zaretti ◽  
Paolo Zunino
Keyword(s):  
Weak Formulation ◽  
Evolution Problems

scholarly journals A derivation of the Liouville equation for hard particle dynamics with non-conservative interactions

2020 ◽  
pp. 1-35
Author(s):  
Benjamin D. Goddard ◽  
Tim D. Hurst ◽  
Mark Wilkinson
Keyword(s):  
Liouville Equation ◽  
Particle Dynamics ◽  
Fundamental Importance ◽  
Continuum Models ◽  
Hard Particle ◽  
Interacting Particles ◽  
Weak Formulation ◽  
Physical Systems ◽  
Physical Particle

The Liouville equation is of fundamental importance in the derivation of continuum models for physical systems which are approximated by interacting particles. However, when particles undergo instantaneous interactions such as collisions, the derivation of the Liouville equation must be adapted to exclude non-physical particle positions, and include the effect of instantaneous interactions. We present the weak formulation of the Liouville equation for interacting particles with general particle dynamics and interactions, and discuss the results using two examples.

Geometrically exact, intrinsic mixed variational formulation for smart, slender multilink composite structures

2021 ◽  
pp. 1045389X2110322
Author(s):  
P M G Bashir Asdaque ◽  
Sitikantha Roy
Keyword(s):  
Composite Structures ◽  
Turbine Blades ◽  
Space Structures ◽  
Wind Turbine Blades ◽  
Weak Formulation ◽  
Helicopter Blades ◽  
Static Mechanical ◽  
Computational Platform ◽  
Swept Wings ◽  
Variational Statement

Flexible links are often part of massive aerospace structures like helicopter or wind turbine blades, satellite bae, airplane wings, and space stations. In the present work, a mixed variational statement based on intrinsic variables is derived for multilinked smart slender structures. Equations involved in the derivation do not involve approximations of kinematical variables to describe the deformation of the reference line or the rotation of the deformed cross-section of the slender links resulting in a geometrically exact formulation. Finite element equations are derived from weak formulation, which can analyze large geometrically non-linear problems. The weakest possible variational statement provides greater flexibility in the choice of shape functions, therefore reducing the associated numerical complexities. The present work focuses on developing a single integrated computational platform which can study multibody, multilink, lightweight composite, structural system built with both embedded actuations, sensing, as well as passive links. Validation of static mechanical and electrical outputs from 3D FE simulation and literature proves the efficacy of the computational platform. Dynamic results will be communicated in future correspondence. The computational platform developed here can be applied for monitoring and active control applications of flexible smart multilink structures like swept wings, multi-bae space structures, and helicopter blades.

scholarly journals Non-Isothermal Creeping Flows in a Pipeline Network: Existence Results

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1300
Author(s):  
Evgenii S. Baranovskii ◽  
Vyacheslav V. Provotorov ◽  
Mikhail A. Artemov ◽  
Alexey P. Zhabko
Keyword(s):  
Nonlinear Boundary ◽  
Galerkin Approximation ◽  
Large Data ◽  
Temperature Dependent ◽  
Weak Formulation ◽  
Temperature Dependent Viscosity ◽  
Pipeline Network ◽  
Flux Boundary Conditions ◽  
Creeping Flows ◽  
Dependent Viscosity

This paper deals with a 3D mathematical model for the non-isothermal steady-state flow of an incompressible fluid with temperature-dependent viscosity in a pipeline network. Using the pressure and heat flux boundary conditions, as well as the conjugation conditions to satisfy the mass balance in interior junctions of the network, we propose the weak formulation of the nonlinear boundary value problem that arises in the framework of this model. The main result of our work is an existence theorem (in the class of weak solutions) for large data. The proof of this theorem is based on a combination of the Galerkin approximation scheme with one result from the field of topological degrees for odd mappings defined on symmetric domains.

scholarly journals Regularity and time behavior of the solutions to weak monotone parabolic equations

2021 ◽  
Author(s):  
Maria Michaela Porzio
Keyword(s):  
Heat Equation ◽  
Parabolic Equations ◽  
Parabolic Problems ◽  
Summable Function ◽  
Time Behavior ◽  
Decay Estimates ◽  
Laplacian Equation ◽  
Evolution Problems ◽  
Degenerate Equations ◽  
Uniqueness Results

AbstractIn this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum $$u_0$$ u 0 is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of $$u_0$$ u 0 , immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems.

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