The fictitious domain method with penalty for the parabolic problem in moving-boundary domain: The error estimate of penalty and the finite element approximation

The fictitious domain technique is coupled to the improved time-explicit asymptotic method for calculating time-periodic solution of wave equation. Conventionally, the practical implementation of fictitious domain method relies on finite difference time discretizations schemes and finite element approximation. Our new method applies finite difference approximations in space instead of conventional finite element approximation. We use the Dirac delta function to transport the variational forms of the wave equations to the differential form and then solve it by finite difference schemes. Our method is relatively easier to code and requires fewer computational operations than conventional finite element method. The numerical experiments show that the new method performs as well as the method using conventional finite element approximation.

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Numerical Analysis for a Nonlocal Parabolic Problem

East Asian Journal on Applied Mathematics ◽

10.4208/eajam.260516.150816a ◽

2016 ◽

Vol 6 (4) ◽

pp. 434-447 ◽

Cited By ~ 3

Author(s):

M. Mbehou ◽

R. Maritz ◽

P.M.D. Tchepmo

Keyword(s):

Finite Element ◽

Parabolic Problem ◽

Reaction Diffusion ◽

Finite Element Approximation ◽

Reaction Diffusion Equation ◽

Element Approximation ◽

Galerkin Finite Element ◽

Fully Discrete ◽

Nonlinear Reaction Diffusion Equation ◽

Nonlinear Parabolic

AbstractThis article is devoted to the study of the finite element approximation for a nonlocal nonlinear parabolic problem. Using a linearised Crank-Nicolson Galerkin finite element method for a nonlinear reaction-diffusion equation, we establish the convergence and error bound for the fully discrete scheme. Moreover, important results on exponential decay and vanishing of the solutions in finite time are presented. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

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Finite element approximation of a nonlinear parabolic problem

Computers & Mathematics with Applications ◽

10.1016/0898-1221(78)90036-6 ◽

1978 ◽

Vol 4 (3) ◽

pp. 247-255 ◽

Cited By ~ 8

Author(s):

B. Neta

Keyword(s):

Finite Element ◽

Parabolic Problem ◽

Finite Element Approximation ◽

Element Approximation ◽

Nonlinear Parabolic Problem ◽

Nonlinear Parabolic

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Mesh Quality and More Detailed Error Estimates of Finite Element Method

Numerical Mathematics Theory Methods and Applications ◽

10.4208/nmtma.2017.s10 ◽

2017 ◽

Vol 10 (2) ◽

pp. 420-436

Author(s):

Yunqing Huang ◽

Liupeng Wang ◽

Nianyu Yi

Keyword(s):

Finite Element ◽

Error Estimate ◽

Error Estimates ◽

A Priori ◽

Finite Element Approximation ◽

Element Approximation ◽

Mesh Quality ◽

Yield Information ◽

Symmetry Structure

AbstractIn this paper, we study the role of mesh quality on the accuracy of linear finite element approximation. We derive a more detailed error estimate, which shows explicitly how the shape and size of elements, and symmetry structure of mesh effect on the error of numerical approximation. Two computable parameters Ge and Gv are given to depict the cell geometry property and symmetry structure of the mesh. In compare with the standard a priori error estimates, which only yield information on the asymptotic error behaviour in a global sense, our proposed error estimate considers the effect of local element geometry properties, and is thus more accurate. Under certain conditions, the traditional error estimates and supercovergence results can be derived from the proposed error estimate. Moreover, the estimators Ge and Gv are computable and thus can be used for predicting the variation of errors. Numerical tests are presented to illustrate the performance of the proposed parameters Ge and Gv.

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Improving the Accuracy of Fictitious Domain Method Using Indicator Function from Volume Intersection

Advances in Mathematical Physics ◽

10.1155/2019/5450313 ◽

2019 ◽

Vol 2019 ◽

pp. 1-18

Author(s):

Guoliang Chai ◽

Junwei Su ◽

Le Wang ◽

Chunlei Yu ◽

Yigen Zhang ◽

...

Keyword(s):

Circular Cylinder ◽

Boundary Problem ◽

Moving Boundary ◽

Indicator Function ◽

Fictitious Domain ◽

Moving Boundary Problems ◽

Fictitious Domain Method ◽

Boundary Problems ◽

Domain Method ◽

Volume Intersection

Fictitious domain method (FDM) is a commonly accepted direct numerical simulation technique for moving boundary problems. Indicator function used to distinguish the solid zone and the fluid zone is an essential part concerning the whole prediction accuracy of FDM. In this work, a new indicator function through volume intersection is developed for FDM. In this method, the arbitrarily polyhedral cells across the interface between fluid and solid are located and subdivided into tetrahedrons. The fraction of the solid volume in each cell is accurately computed to achieve high precision of integration calculation in the particle domain, improving the accuracy of the whole method. The quadrature over the solid domain shows that the newly developed indicator function can provide results with high accuracy for variable integration in both stationary and moving boundary problems. Several numerical tests, including flow around a circular cylinder, a single sphere in a creeping shear flow, settlement of a circular particle in a closed container, and in-line oscillation of a circular cylinder, have been performed. The results show good accuracy and feasibility in dealing with the stationary boundary problem as well as the moving boundary problem. This method is accurate and conservative, which can be a feasible tool for studying problems with moving boundaries.

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THE TECHNIQUE OF THE IMMERSED BOUNDARY METHOD: APPLICATIONS TO THE NUMERICAL SOLUTIONS OF IMCOMPRESSIBLE FLOWS AND WAVE SCATTERING

Modern Physics Letters B ◽

10.1142/s021798490901859x ◽

2009 ◽

Vol 23 (03) ◽

pp. 437-440 ◽

Cited By ~ 1

Author(s):

HONGQUAN CHEN ◽

LING RAO

Keyword(s):

Numerical Simulation ◽

Finite Element ◽

Immersed Boundary Method ◽

Fictitious Domain ◽

Immersed Boundary ◽

Fictitious Domain Method ◽

Lagrangian Multipliers ◽

Element Discretization ◽

Domain Method ◽

Boundary Method

A fictitious domain method, in which the Dirichlet boundary conditions are treated using boundary supported Lagrangian multipliers, is considered. The technique of the immersed boundary method is incorporated into the framework of the fictitious domain method. Contrary to conventional methods, it does not make use of the finite element discretization. It has a simpler structure and is easily programmable. The numerical simulation of two-dimensional incompressible inviscid uniform flows over a circular cylinder validates the methodology and the numerical procedure. The numerical simulation of propagation phenomena for time harmonic electromagnetic waves by methods combining controllability and fictitious domain techniques is also presented. Using distributed Lagrangian multipliers, the propagation of the wave can be simulated on an obstacle free computational region with regular finite element meshes essentially independent of the geometry of the obstacle and by a controllability formulation which leads to algorithms with good convergence properties for time-periodic solutions. The numerical results presented are in good agreement with those in the literature using obstacle fitted meshes.

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