scholarly journals Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

2011 ◽  
Vol 10 (3) ◽  
pp. 509-576 ◽  
Author(s):  
M. J. Baines ◽  
M. E. Hubbard ◽  
P. K. Jimack
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Historical Review ◽  
Porous Medium Equation ◽  
Moving Mesh ◽  
Test Problems ◽  
Partial Differential ◽  
Two Phase

AbstractThis article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

scholarly journals A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1336
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu ◽  
Dumitru Ţucu ◽  
Marioara Lăpădat ◽  
Mădălina Sofia Paşca
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Least Squares ◽  
Least Squares Method ◽  
Differential Quadrature Method ◽  
Nonlinear Partial Differential Equations ◽  
Differential Quadrature ◽  
Test Problems ◽  
Quadrature Method ◽  
Partial Differential

In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.

Continuous Solution of Partial Differential Equations on Non-Rectangular and Non-Continuous Geometry

2014 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Continuous Solution ◽  
Finite Difference Methods ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Mesh Free ◽  
Numeric Calculation ◽  
Linear And Nonlinear

A high order continuous solution is obtained for partial differential equations on non-rectangular and non-continuous domain using Bézier functions. This is a mesh free alternative to finite element or finite difference methods that are normally used to solve such problems. The problem is handled without any transformation and the setup is direct, simple, and involves minimizing the error in the residuals of the differential equations along with the error in the boundary conditions over the domain. The solution can be expressed in polynomial form. The effort is same for linear and nonlinear partial differential equations. The procedure is developed as a combination of symbolic and numeric calculation. The solution is obtained through the application of standard unconstrained optimization. A constrained approach is also developed for nonlinear partial differential equations. Examples include linear and nonlinear partial differential equations. The solution for linear partial differential equations is compared to finite element solutions from COMSOL.

scholarly journals A numerical method using Laplace-like transform and variational theory for solving time-fractional nonlinear partial differential equations with proportional delay

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Alemu Senbeta Bekela ◽  
Melisew Tefera Belachew ◽  
Getinet Alemayehu Wole
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Nonlinear Partial Differential Equations ◽  
Stability And Convergence ◽  
Test Problems ◽  
Proportional Delay ◽  
Variational Theory ◽  
Partial Differential ◽  
The Stability

Abstract Time-fractional nonlinear partial differential equations (TFNPDEs) with proportional delay are commonly used for modeling real-world phenomena like earthquake, volcanic eruption, and brain tumor dynamics. These problems are quite challenging, and the transcendental nature of the delay makes them even more difficult. Hence, the development of efficient numerical methods is open for research. In this paper, we use the concepts of Laplace-like transform and variational theory to develop a new numerical method for solving TFNPDEs with proportional delay. The stability and convergence of the method are analyzed in the Banach sense. The efficiency of the proposed method is demonstrated by solving some test problems. The numerical results show that the proposed method performs much better than some recently developed methods and enables us to obtain more accurate solutions.

scholarly journals Finite element approximation of an incompressible chemically reacting non-Newtonian fluid

2018 ◽  
Vol 52 (2) ◽  
pp. 509-541 ◽  
Author(s):  
Seungchan Ko ◽  
Petra Pustějovská ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Newtonian Fluid ◽  
Power Law ◽  
Nonlinear Partial Differential Equations ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Partial Differential ◽  
Chemically Reacting

We consider a system of nonlinear partial differential equations modelling the steady motion of an incompressible non-Newtonian fluid, which is chemically reacting. The governing system consists of a steady convection-diffusion equation for the concentration and the generalized steady Navier–Stokes equations, where the viscosity coefficient is a power-law type function of the shear-rate, and the coupling between the equations results from the concentration-dependence of the power-law index. This system of nonlinear partial differential equations arises in mathematical models of the synovial fluid found in the cavities of moving joints. We construct a finite element approximation of the model and perform the mathematical analysis of the numerical method in the case of two space dimensions. Key technical tools include discrete counterparts of the Bogovskiĭ operator, De Giorgi’s regularity theorem in two dimensions, and the Acerbi–Fusco Lipschitz truncation of Sobolev functions, in function spaces with variable integrability exponents.

scholarly journals Finite element approximation of steady flows of colloidal solutions

2021 ◽  
Author(s):  
Andrea Bonito ◽  
Vivette Girault ◽  
Diane Guignard ◽  
Kumbakonam R. Rajagopal ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Weak Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Cauchy Stress ◽  
Partial Differential ◽  
Fixed Point Algorithm

We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to  steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions-Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.

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