scholarly journals On nonlinear problems of parabolic type with implicit constitutive equations involving flux

2021 ◽  
pp. 1-52
Author(s):  
Miroslav Bulíček ◽  
Erika Maringová ◽  
Josef Málek
Keyword(s):  
Nonlinear Partial Differential Equations ◽  
Maximal Monotone ◽  
Parabolic Type ◽  
Nonlinear Problems ◽  
Large Data ◽  
Parabolic Problems ◽  
Monotone Mappings ◽  
Spatial Gradient ◽  
Implicit Equation ◽  
Monotone Formula

We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first-order divergence operator acting on a flux function, which is related to the spatial gradient of the unknown through an additional implicit equation. This setting, broad enough in terms of applications, significantly expands the paradigm of nonlinear parabolic problems. Formulating four conditions concerning the form of the implicit equation, we first show that these conditions describe a maximal monotone [Formula: see text]-coercive graph. We then establish the global-in-time and large-data existence of a (weak) solution and its uniqueness. To this end, we adopt and significantly generalize Minty’s method of monotone mappings. A unified theory, containing several novel tools, is developed in a way to be tractable from the point of view of numerical approximations.

Explicit algorithms for J-fixed points of some non linear mappings in certain Banach spaces

2020 ◽  
Vol 29 (1) ◽  
pp. 27-36
Author(s):  
M. M. GUEYE ◽  
M. SENE ◽  
M. NDIAYE ◽  
N. DJITTE
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Maximal Monotone ◽  
Point Theory ◽  
Duality Mapping ◽  
Monotone Mappings ◽  
Normalized Duality Mapping ◽  
Maximal Monotone Mappings ◽  
Pseudocontractive Mappings

Let E be a real normed linear space and E∗ its dual. In a recent work, Chidume et al. [Chidume, C. E. and Idu, K. O., Approximation of zeros of bounded maximal monotone mappings, solutions of hammerstein integral equations and convex minimizations problems, Fixed Point Theory and Applications, 97 (2016)] introduced the new concepts of J-fixed points and J-pseudocontractive mappings and they shown that a mapping A : E → 2 E∗ is monotone if and only if the map T := (J −A) : E → 2 E∗ is J-pseudocontractive, where J is the normalized duality mapping of E. It is our purpose in this work to introduce an algorithm for approximating J-fixed points of J-pseudocontractive mappings. Our results are applied to approximate zeros of monotone mappings in certain Banach spaces. The results obtained here, extend and unify some recent results in this direction for the class of maximal monotone mappings in uniformly smooth and strictly convex real Banach spaces. Our proof is of independent interest.

scholarly journals On unsteady flows of pore pressure-activated granular materials

2020 ◽  
Vol 72 (1) ◽  
Author(s):  
Anna Abbatiello ◽  
Miroslav Bulíček ◽  
Tomáš Los ◽  
Josef Málek ◽  
Ondřej Souček
Keyword(s):  
Granular Materials ◽  
Plastic Material ◽  
Pore Space ◽  
Nonlinear Partial Differential Equations ◽  
Large Data ◽  
Existence Theory ◽  
Solid Matrix ◽  
Lithostatic Pressure ◽  
Generalized Newtonian Fluid ◽  
Stick Slip

AbstractWe investigate mathematical properties of the system of nonlinear partial differential equations that describe, under certain simplifying assumptions, evolutionary processes in water-saturated granular materials. The unconsolidated solid matrix behaves as an ideal plastic material before the activation takes place and then it starts to flow as a Newtonian or a generalized Newtonian fluid. The plastic yield stress is non-constant and depends on the difference between the given lithostatic pressure and the pressure of the fluid in a pore space. We study unsteady three-dimensional flows in an impermeable container, subject to stick-slip boundary conditions. Under realistic assumptions on the data, we establish long-time and large-data existence theory.

scholarly journals Strong Convergence Theorems for Quasi-Bregman Nonexpansive Mappings in Reflexive Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Mohammed Ali Alghamdi ◽  
Naseer Shahzad ◽  
Habtu Zegeye
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Finite Family ◽  
Monotone Mappings ◽  
Reflexive Banach Spaces ◽  
Convergence Theorems

We study a strong convergence for a common fixed point of a finite family of quasi-Bregman nonexpansive mappings in the framework of real reflexive Banach spaces. As a consequence, convergence for a common fixed point of a finite family of Bergman relatively nonexpansive mappings is discussed. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding a common solution of a finite family equilibrium problem and a common zero of a finite family of maximal monotone mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

scholarly journals Superconvergence analysis of flux computations for nonlinear problems

1989 ◽  
Vol 40 (3) ◽  
pp. 465-479 ◽  
Author(s):  
S.-S. Chow ◽  
R.D. Lazarov
Keyword(s):  
Elliptic Boundary ◽  
Nonlinear Problems ◽  
Parabolic Problems ◽  
Elliptic Boundary Value ◽  
Strongly Nonlinear ◽  
Nonlinear Elliptic ◽  
General Line ◽  
Nonlinear Elliptic Boundary ◽  
Boundary Flux ◽  
Semilinear Parabolic

In this paper we consider the error estimates for some boundary-flux calculation procedures applied to two-point semilinear and strongly nonlinear elliptic boundary value problems. The case of semilinear parabolic problems is also studied. We prove that the computed flux is superconvergent with second and third order of convergence for linear and quadratic elements respectively. Corresponding estimates for higher order elements may also be obtained by following the general line of argument.

scholarly journals Development of a parallel algorithm based on an implicit scheme for the discontinuous Galerkin method for solving diffusion type equations

2020 ◽  
Vol 22 (1) ◽  
pp. 94-106
Author(s):  
Ruslan V. Zhalnin ◽  
Nikita A. Kuzmin ◽  
Victor F. Masyagin
Keyword(s):  
Galerkin Method ◽  
Parallel Algorithm ◽  
Numerical Study ◽  
Sparse Matrices ◽  
Parabolic Type ◽  
Implicit Scheme ◽  
Basis Functions ◽  
Parabolic Problems ◽  
Diffusion Type ◽  
The Galerkin Method

The paper presents a numerical parallel algorithm based on an implicit scheme for the Galerkin method with discontinuous basis functions for solving diffusion-type equations on triangular grids. To apply the Galerkin method with discontinuous basis functions, the initial equation of parabolic type is transformed to a system of partial differential equations of the first order. To do this, auxiliary variables are introduced, which are the components of the gradient of the desired function. To store sparse matrices and vectors, the CSR format is used in this study. The resulting system is solved numerically using a parallel algorithm based on the Nvidia AmgX library. A numerical study is carried out on the example of solving two-dimensional test parabolic initial-boundary value problems. The presented numerical results show the effectiveness of the proposed algorithm for solving parabolic problems.

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