scholarly journals Error estimates for the semidiscrete finite element approximation of linear nonlocal parabolic equations

1992 ◽  
Vol 5 (1) ◽  
pp. 19-27 ◽  
Author(s):  
Dennis E. Jackson
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Finite Element Approximation ◽  
Parabolic Partial Differential Equations ◽  
Element Approximation ◽  
Initial Condition ◽  
Initial Value ◽  
Semidiscrete Approximation ◽  
Nonlocal Parabolic Equations

Existence and uniqueness are proved for nonlocal (in time) for solutions of linear parabolic partial differential equations. Instead of an initial condition, there is a relation connecting the initial value to values of the solution at other times. L2 error estimates are obtained for the semidiscrete approximation of the problem using finite elements in the space variables.

scholarly journals EXISTENCE, UNIQUENESS AND APPROXIMATION OF A DOUBLY-DEGENERATE NONLINEAR PARABOLIC SYSTEM MODELLING BACTERIAL EVOLUTION

2007 ◽  
Vol 17 (07) ◽  
pp. 1095-1127 ◽  
Author(s):  
JOHN W. BARRETT ◽  
KLAUS DECKELNICK
Keyword(s):  
Parabolic System ◽  
Existence And Uniqueness ◽  
Finite Element Approximation ◽  
System Modelling ◽  
Element Approximation ◽  
Nonlinear Parabolic System ◽  
Spatiotemporal Evolution ◽  
Nonlinear Parabolic ◽  
Flux Boundary Conditions ◽  
Bacterium Species

We consider the following nonlinear parabolic system [Formula: see text] subject to no flux boundary conditions, and non-negative initial data u0 and v0 on u and v. Here we assume that c > 0, θ ≥ 0 and that [Formula: see text] is increasing with f(0) = 0. The system is possibly doubly-degenerate in that [Formula: see text] is only non-negative, and ψ ∈ C1([0,∞)) ∩ C2((0,∞)) is convex, strictly increasing with ψ(0) = 0 and possibly ψ'(0) = 0. The above models the spatiotemporal evolution of a bacterium species on a thin film of nutrient, where u is the nutrient concentration and v is the bacterial cell density. Under some further mild technical assumptions on b and ψ, we prove the existence and uniqueness of a weak solution to the above system. Moreover, we prove error bounds for a fully practical finite element approximation of this system. All of our results apply to the choices b(r) ≔ rq and ψ(r) ≔ rp with q ≥ 2 and p ≥ 1, for example.

scholarly journals Well-posedness and H(div)-conforming finite element approximation of a linearised model for inviscid incompressible flow

2020 ◽  
Vol 30 (05) ◽  
pp. 847-865
Author(s):  
Gabriel Barrenechea ◽  
Erik Burman ◽  
Johnny Guzmán
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Model Problem ◽  
Finite Element Approximation ◽  
Inviscid Flow ◽  
Element Approximation ◽  
Well Posedness ◽  
Hodge Laplacian ◽  
Conforming Finite Element ◽  
Velocity Approximation

We consider a linearised model of incompressible inviscid flow. Using a regularisation based on the Hodge Laplacian we prove existence and uniqueness of weak solutions for smooth domains. The model problem is then discretised using [Formula: see text](div)-conforming finite element methods, for which we prove error estimates for the velocity approximation in the [Formula: see text]-norm of order [Formula: see text]. We also prove error estimates for the pressure error in the [Formula: see text]-norm.

Composite Finite Element Approximation for Parabolic Problems in Nonconvex Polygonal Domains

2020 ◽  
Vol 20 (2) ◽  
pp. 361-378
Author(s):  
Tamal Pramanick ◽  
Rajen Kumar Sinha
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Error Estimates ◽  
Finite Element Approximation ◽  
Parabolic Problems ◽  
Element Approximation ◽  
Dirichlet Problems ◽  
Element Mesh ◽  
Polygonal Domains ◽  
Numer Math

AbstractThe purpose of this paper is to generalize known a priori error estimates of the composite finite element (CFE) approximations of elliptic problems in nonconvex polygonal domains to the time dependent parabolic problems. This is a new class of finite elements which was introduced by [W. Hackbusch and S. A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math. 75 1997, 4, 447–472] and subsequently modified by [M. Rech, S. A. Sauter and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Numer. Math. 102 2006, 4, 681–708] for the approximations of stationery problems on complicated domains. The basic idea of the CFE procedure is to work with fewer degrees of freedom by allowing finite element mesh to resolve the domain boundaries and to preserve the asymptotic order convergence on coarse-scale mesh. We analyze both semidiscrete and fully discrete CFE methods for parabolic problems in two-dimensional nonconvex polygonal domains and derive error estimates of order {\mathcal{O}(H^{s}\widehat{\mathrm{Log}}{}^{\frac{s}{2}}(\frac{H}{h}))} and {\mathcal{O}(H^{2s}\widehat{\mathrm{Log}}{}^{s}(\frac{H}{h}))} in the {L^{\infty}(H^{1})}-norm and {L^{\infty}(L^{2})}-norm, respectively. Moreover, for homogeneous equations, error estimates are derived for nonsmooth initial data. Numerical results are presented to support the theoretical rates of convergence.

scholarly journals Existence and uniqueness of analytic solutions of the Shabat equation

2005 ◽  
Vol 2005 (8) ◽  
pp. 855-862 ◽  
Author(s):  
Eugenia N. Petropoulou
Keyword(s):  
Upper Bound ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Analytic Solution ◽  
Sufficient Conditions ◽  
Analytic Solutions ◽  
Initial Condition ◽  
Initial Value ◽  
First Derivative

Sufficient conditions are given so that the initial value problem for the Shabat equation has a unique analytic solution, which, together with its first derivative, converges absolutely forz∈ℂ:|z|<T,T>0. Moreover, a bound of this solution is given. The sufficient conditions involve only the initial condition, the parameters of the equation, andT. Furthermore, from these conditions, one can obtain an upper bound forT. Our results are in consistence with some recently found results.

A prioriestimates and optimal finite element approximation of the MHD flow in smooth domains

2018 ◽  
Vol 52 (1) ◽  
pp. 181-206 ◽  
Author(s):  
Yinnian He ◽  
Jun Zou
Keyword(s):  
Magnetic Field ◽  
Finite Element ◽  
Error Estimates ◽  
Mhd Flow ◽  
Finite Element Approximation ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Optimal Error ◽  
Discrete Velocity ◽  
Mhd System

We study a finite element approximation of the initial-boundary value problem of the 3D incompressible magnetohydrodynamic (MHD) system under smooth domains and data. We first establish several important regularities anda prioriestimates for the velocity, pressure and magnetic field (u,p,B) of the MHD system under the assumption that ∇u∈L4(0,T;L2(Ω)3 × 3) and ∇ ×B∈L4(0,T;L2(Ω)3). Then we formulate a finite element approximation of the MHD flow. Finally, we derive the optimal error estimates of the discrete velocity and magnetic field in energy-norm and the discrete pressure inL2-norm, and the optimal error estimates of the discrete velocity and magnetic field inL2-norm by means of a novel negative-norm technique, without the help of the standard duality argument for the Navier-Stokes equations.

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