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.

scholarly journals Goal-Oriented Error Estimation for the Fractional Step Theta Scheme

2014 ◽  
Vol 14 (2) ◽  
pp. 203-230 ◽  
Author(s):  
Dominik Meidner ◽  
Thomas Richter
Keyword(s):  
Time Discretization ◽  
Posteriori Error ◽  
Numerical Dissipation ◽  
Error Estimator ◽  
Fractional Step ◽  
Weighted Residual Method ◽  
Time Stepping ◽  
Backward Euler ◽  
Galerkin Formulation ◽  
Theta Method

Abstract. In this work, we derive a goal-oriented a posteriori error estimator for the error due to time-discretization of nonlinear parabolic partial differential equations by the fractional step theta method. This time-stepping scheme is assembled by three steps of the general theta method, that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson method. Further, by combining three substeps of the theta time-stepping scheme, the fractional step theta time-stepping scheme is derived. It possesses highly desired stability and numerical dissipation properties and is second order accurate. The derived error estimator is based on a Petrov–Galerkin formulation that is up to a numerical quadrature error equivalent to the theta time-stepping scheme. The error estimator is assembled as one weighted residual term given by the dual weighted residual method and one additional residual estimating the Galerkin error between time-stepping scheme and Petrov–Galerkin formulation.

scholarly journals Convergence theorems for generalized projections and maximal monotone operators in Banach spaces

2003 ◽  
Vol 2003 (10) ◽  
pp. 621-629 ◽  
Author(s):  
Takanori Ibaraki ◽  
Yasunori Kimura ◽  
Wataru Takahashi
Keyword(s):  
Banach Space ◽  
Pointwise Convergence ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Convex Banach Space ◽  
Limit Set ◽  
Strong And Weak Convergence ◽  
Strictly Convex Banach Space ◽  
Generalized Projections

We study a sequence of generalized projections in a reflexive, smooth, and strictly convex Banach space. Our result shows that Mosco convergence of their ranges implies their pointwise convergence to the generalized projection onto the limit set. Moreover, using this result, we obtain strong and weak convergence of resolvents for a sequence of maximal monotone operators.

Stable Non-symmetric Coupling of the Finite Volume Method and the Boundary Element Method for Convection-Dominated Parabolic-Elliptic Interface Problems

2020 ◽  
Vol 20 (2) ◽  
pp. 251-272
Author(s):  
Christoph Erath ◽  
Robert Schorr
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Time Discretization ◽  
Euler Method ◽  
Interface Problems ◽  
Backward Euler Method ◽  
Elliptic Interface Problems ◽  
Practical Applications ◽  
Backward Euler ◽  
Volume Method

AbstractMany problems in electrical engineering or fluid mechanics can be modeled by parabolic-elliptic interface problems, where the domain for the exterior elliptic problem might be unbounded. A possibility to solve this class of problems numerically is the non-symmetric coupling of finite elements (FEM) and boundary elements (BEM) analyzed in [H. Egger, C. Erath and R. Schorr, On the nonsymmetric coupling method for parabolic-elliptic interface problems, SIAM J. Numer. Anal. 56 2018, 6, 3510–3533]. If, for example, the interior problem represents a fluid, this method is not appropriate since FEM in general lacks conservation of numerical fluxes and in case of convection dominance also stability. A possible remedy to guarantee both is the use of the vertex-centered finite volume method (FVM) with an upwind stabilization option. Thus, we propose a (non-symmetric) coupling of FVM and BEM for a semi-discretization of the underlying problem. For the subsequent time discretization we introduce two options: a variant of the backward Euler method which allows us to develop an analysis under minimal regularity assumptions and the classical backward Euler method. We analyze both, the semi-discrete and the fully-discrete system, in terms of convergence and error estimates. Some numerical examples illustrate the theoretical findings and give some ideas for practical applications.

scholarly journals Strong and Weak Convergence of Modified Mann Iteration for New Resolvents of Maximal Monotone Operators in Banach Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-20 ◽  
Author(s):  
Somyot Plubtieng ◽  
Wanna Sriprad
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Convergence Rate ◽  
Minimization Problem ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Convex Minimization Problem ◽  
Mann Iteration ◽  
Strong And Weak Convergence

We prove strong and weak convergence theorems for a new resolvent of maximal monotone operators in a Banach space and give an estimate of the convergence rate of the algorithm. Finally, we apply our convergence theorem to the convex minimization problem. The result present in this paper extend and improve the corresponding result of Ibaraki and Takahashi (2007), and Kim and Xu (2005).

Stability and consistency of a finite difference scheme for compressible viscous isentropic flow in multi-dimension

2018 ◽  
Vol 26 (3) ◽  
pp. 111-140 ◽  
Author(s):  
Radim Hošek ◽  
Bangwei She
Keyword(s):  
Time Discretization ◽  
Euler Method ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Numerical Density ◽  
Existence Of A Solution ◽  
Isentropic Flow ◽  
Backward Euler ◽  
Spatial Dimensions ◽  
Stability And Consistency

Abstract Motivated by the work of Karper [29], we propose a numerical scheme to compressible Navier-Stokes system in spatial multi-dimension based on finite differences. The backward Euler method is applied for the time discretization, while a staggered grid, with continuity and momentum equations on different grids, is used in space. The existence of a solution to the implicit nonlinear scheme, strictly positivity of the numerical density, stability and consistency of the method for the whole range of physically relevant adiabatic exponents are proved. The theoretical part is complemented by computational results that are performed in two spatial dimensions.

scholarly journals CONSTRUCCIÓN DE MAPAS DE RUIDO CON ELEMENTOS FINITOS Y NEWMARK

2017 ◽  
Vol 22 (2) ◽  
pp. 35
Author(s):  
Irla Mantilla ◽  
Jonathan Munguia
Keyword(s):  
Finite Element ◽  
Existence And Uniqueness ◽  
Noise Source ◽  
Dimensional Space ◽  
Time Discretization ◽  
Hyperbolic Type ◽  
Newmark Method ◽  
The Finite Element Method ◽  
Weak Formulation ◽  
Palabras Clave

En el presente trabajo se construye un modelo matemático para la construcción de Mapas de Ruido basado en Ecuaciones Variacionales Hiperbólicas (EVP), el cual, se obtiene de la formulación débil del problema de Contorno y Condiciones iniciales de Cauchy asociadas a Ecuaciones Diferenciales en Derivadas Parciales de tipo Hiperbólico. Para garantizar la simulación numérica del problema de propagación de la fuente de ruido se obtiene su formulación variacional en espacios de tipo Sobolev evolutivos, así se prueba la existencia y unicidad de solución del problema variacional, luego para resolver el problema se aplica el método de Galerkin con Elementos Finitos para la discretización espacial y el método de Newmark para la discretización temporal. En este trabajo se innova la técnica de preparación de la base de datos y la experimentación computacional con Matlab, obteniendo finalmente eficazmente la solución como se muestra en la convergencia del esquema numérico. Palabras clave.- Mapas de ruido, Método de Galerkin, Elementos finitos, Método de Newmark. ABSTRACTThis article uses the weak formulation of the problems Partial Hyperbolic type (VPE) for construction noise maps with finite element and Newmark in two-dimensional space. To ensure the numerical simulation of the problem of propagation of the noise source, so that proves the existence and uniqueness of the variational problem in Sobolev spaces evolutionary and applied the finite element method and method Galerkin for spatial discretization and Newmark method for the time discretization. In this work, the innovative technique of preparation of the data base and computational experimentation whit Mathlab, finally obtaining effectively the solution as shown in the convergence of the numerical scheme. Keywords.- Noise maps, Galerkin method, Finite elements, Newmark method.

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