scholarly journals Numerical discretization and fast approximation of a variably distributed-order fractional wave equation

2021 ◽  
Author(s):  
Jinhong Jia ◽  
Xiangcheng Zheng ◽  
Hong Wang
Keyword(s):  
Error Estimates ◽  
Computational Cost ◽  
Viscoelastic Behavior ◽  
Mesh Size ◽  
Optimal Order ◽  
Optimal Error Estimates ◽  
Order Error ◽  
Time Stepping ◽  
Discrete Finite Element ◽  
Distributed Order

We investigate a variably distributed-order time-fractional wave partial differential equation, which could accurately model, e.g., the viscoelastic behavior in vibrations in complex surroundings with uncertainties or strong heterogeneity in the data. A standard composite rectangle formula of mesh size $\sigma$ is firstly used to discretize the variably distributed-order integral and then the L-1 formula of degree of freedom $N$ is applied for the resulting fractional derivatives. Optimal error estimates of the corresponding fully-discrete finite element method are proved based only on the smoothness assumptions of the data. To maintain the accuracy, setting $\sigma=N^{-1}$ leads to $O(N^3)$ operations of evaluating the temporal discretization coefficients and $O(N^2)$ memory. To improve the computational efficiency, we develop a novel time-stepping scheme by expanding the fractional kernel at a fixed fractional order to decouple the fractional operator from the variably distributed-order integral. Only $O(\log N)$ terms are needed for the expansion without loss of accuracy, which consequently reduce the computational cost of coefficients from $O(N^3)$ to $O(N^2\log N)$ and the corresponding memory from $O(N^2)$ to $O(N\log N)$. Optimal-order error estimates of this time-stepping scheme are rigorously proved via novel and different techniques from the standard analysis procedure of the L-1 methods. Numerical experiments are presented to substantiate the theoretical results.

scholarly journals Variational analysis of the discontinuous Galerkin time-stepping method for parabolic equations

2020 ◽  
Author(s):  
Norikazu Saito
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Parabolic Type ◽  
Finite Element Approximation ◽  
Optimal Order ◽  
Element Approximation ◽  
Step Method ◽  
Order Error ◽  
Time Stepping ◽  
Optimal Order Error Estimates

Abstract The discontinuous Galerkin (DG) time-stepping method applied to abstract evolution equation of parabolic type is studied using a variational approach. We establish the inf-sup condition or Babuška–Brezzi condition for the DG bilinear form. Then, a nearly best approximation property and a nearly symmetric error estimate are obtained as corollaries. Moreover, the optimal order error estimates under appropriate regularity assumption on the solution are derived as direct applications of the standard interpolation error estimates. Our method of analysis is new for the DG time-stepping method; it differs from previous works by which the method is formulated as the one-step method. We apply our abstract results to the finite element approximation of a second-order parabolic equation with space-time variable coefficient functions in a polyhedral domain, and derive the optimal order error estimates in several norms.

scholarly journals Analysis of a Decoupled Time-Stepping Scheme for Evolutionary Micropolar Fluid Flows

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
S. S. Ravindran
Keyword(s):  
Micropolar Fluid ◽  
Stokes Equations ◽  
Fluid Model ◽  
Navier Stokes ◽  
Optimal Order ◽  
Step Size ◽  
Time Step ◽  
Order Error ◽  
Time Step Size ◽  
Time Stepping

Micropolar fluid model consists of Navier-Stokes equations and microrotational velocity equations describing the dynamics of flows in which microstructure of fluid is important. In this paper, we propose and analyze a decoupled time-stepping algorithm for the evolutionary micropolar flow. The proposed method requires solving only one uncoupled Navier-Stokes and one microrotation subphysics problem per time step. We derive optimal order error estimates in suitable norms without assuming any stability condition or time step size restriction.

scholarly journals HP DISCONTINUOUS GALERKIN APPROXIMATIONS FOR THE STOKES PROBLEM

2002 ◽  
Vol 12 (11) ◽  
pp. 1565-1597 ◽  
Author(s):  
ANDREA TOSELLI
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Order Approximation ◽  
Stokes Problem ◽  
Galerkin Approximation ◽  
Mesh Size ◽  
Numerical Results ◽  
Optimal Error Estimates ◽  
Polynomial Degree ◽  
Uniformly Stable

We propose and analyze a discontinuous Galerkin approximation for the Stokes problem. The finite element triangulation employed is not required to be conforming and we use discontinuous pressures and velocities. No additional unknown fields need to be introduced, but only suitable bilinear forms defined on the interfaces between the elements, involving the jumps of the velocity and the average of the pressure. We consider hp approximations using ℚk′–ℚk velocity-pressure pairs with k′ = k + 2, k + 1, k. Our methods show better stability properties than the corresponding conforming ones. We prove that our first two choices of velocity spaces ensure uniform divergence stability with respect to the mesh size h. Numerical results show that they are uniformly stable with respect to the local polynomial degree k, a property that has no analog in the conforming case. An explicit bound in k which is not sharp is also proven. Numerical results show that if equal order approximation is chosen for the velocity and pressure, no spurious pressure modes are present but the method is not uniformly stable either with respect to h or k. We derive a priori error estimates generalizing the abstract theory of mixed methods. Optimal error estimates in h are proven. As for discontinuous Galerkin methods for scalar diffusive problems, half of the power of k is lost for p and hp pproximations independently of the divergence stability.

scholarly journals Analysis of finite element methods for vector Laplacians on surfaces

2019 ◽  
Vol 40 (3) ◽  
pp. 1652-1701 ◽  
Author(s):  
Peter Hansbo ◽  
Mats G Larson ◽  
Karl Larsson
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Vector Fields ◽  
Optimal Order ◽  
Order Error ◽  
Optimal Order Error Estimates ◽  
Elastic Membranes ◽  
Tangential Vector ◽  
Higher Order Finite Elements ◽  
Lagrange Elements

Abstract We develop a finite element method for the vector Laplacian based on the covariant derivative of tangential vector fields on surfaces embedded in ${\mathbb{R}}^3$. Closely related operators arise in models of flow on surfaces as well as elastic membranes and shells. The method is based on standard continuous parametric Lagrange elements that describe a ${\mathbb{R}}^3$ vector field on the surface, and the tangent condition is weakly enforced using a penalization term. We derive error estimates that take into account the approximation of both the geometry of the surface and the solution to the partial differential equation. In particular, we note that to achieve optimal order error estimates, in both energy and $L^2$ norms, the normal approximation used in the penalization term must be of the same order as the approximation of the solution. This can be fulfilled either by using an improved normal in the penalization term, or by increasing the order of the geometry approximation. We also present numerical results using higher-order finite elements that verify our theoretical findings.

scholarly journals Analysis of a Linear–Linear Finite Element for the Reissner–Mindlin Plate Model

1997 ◽  
Vol 07 (02) ◽  
pp. 217-238 ◽  
Author(s):  
Douglas N. Arnold ◽  
Richard S. Falk
Keyword(s):  
Finite Element ◽  
Plate Thickness ◽  
Mesh Size ◽  
Mindlin Plate ◽  
Optimal Order ◽  
Plate Model ◽  
Order Error ◽  
Linear Finite Element ◽  
Optimal Order Error Estimates ◽  
Linear Elements

An analysis is presented for a recently proposed finite element method for the Reissner–Mindlin plate problem. The method is based on the standard variational principle, uses nonconforming linear elements to approximate the rotations and conforming linear elements to approximate the transverse displacements, and avoids the usual "locking problem" by interpolating the shear stress into a rotated space of lowest order Raviart-Thomas elements. When the plate thickness t = O(h), it is proved that the method gives optimal order error estimates uniform in t. However, the analysis suggests and numerical calculations confirm that the method can produce poor approximations for moderate sized values of the plate thickness. Indeed, for t fixed, the method does not converge as the mesh size h tends to zero.

scholarly journals Mixed methods for degenerate elliptic problems and application to fractional Laplacian

2020 ◽  
Author(s):  
María Eugenia Cejas ◽  
Ricardo Durán ◽  
Mariana Prieto
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Mixed Finite Element ◽  
Optimal Order ◽  
Order Error ◽  
Quadrilateral Elements ◽  
Muckenhoupt Class ◽  
Degenerate Elliptic ◽  
Weighted Norms ◽  
Solutions Of Equations

  We analyze the approximation by mixed finite element methods of solutions of     equations of the form div  [[EQUATION]]  , where the coefficient a=a(x) can     degenerate going to zero or infinity. First, we extend the classic error analysis to this case provided that the     coefficient $a$ belongs to the Muckenhoupt class  [[EQUATION]] .     The analysis developed applies to general mixed finite element spaces satisfying the     standard commutative diagram property, whenever some stability and interpolation     error estimates are valid in weighted norms. Next, we consider in detail the case     of Raviart-Thomas spaces of arbitrary order, obtaining optimal order error estimates for simplicial elements in any dimension and for convex quadrilateral elements in the two dimensional case, in both cases under a regularity assumption on the family of meshes.          For the lowest order case we show that the regularity assumprtion can be removed and prove  anisotropic error estimates which are of interest in problems with boundary layers. Finally we apply the results to a problem arising in the solution of the fractional Laplace equation.

scholarly journals A decoupled Crank-Nicolson time-stepping scheme for thermally coupled magneto-hydrodynamic system

2017 ◽  
Vol 8 (1) ◽  
pp. 43-62
Author(s):  
S. S. Ravindran
Keyword(s):  
Heat Equation ◽  
Mixed Finite Element ◽  
Mhd Equations ◽  
Optimal Order ◽  
Step Size ◽  
Time Step ◽  
Order Error ◽  
Time Step Size ◽  
Time Stepping ◽  
Crank Nicolson

Thermally coupled magneto-hydrodynamics (MHD) studies the dynamics of electro-magnetically and thermally driven flows,involving MHD equations coupled with heat equation. We introduce a partitioned method that allows one to decouplethe MHD equations from the heat equation at each time step and solve them separately. The extrapolated Crank-Nicolson time-stepping scheme is used for time discretizationwhile mixed finite element method is used for spatial discretization. We derive optimal order error estimates in suitable norms without assuming any stability condition or restrictions on the time step size. We prove the unconditional stability of the scheme. Numerical experiments are used to illustrate the theoretical results.

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