scholarly journals A Compact FEM Implementation for Parabolic Integro-Differential Equations in 2D

Algorithms ◽  
2020 ◽  
Vol 13 (10) ◽  
pp. 242
Author(s):  
Gujji Murali Mohan Reddy ◽  
Alan B. Seitenfuss ◽  
Débora de Oliveira Medeiros ◽  
Luca Meacci ◽  
Milton Assunção ◽  
...  
Keyword(s):  
Finite Element ◽  
Differential Equations ◽  
Piecewise Linear ◽  
Time Discretization ◽  
Comsol Multiphysics ◽  
Integral Term ◽  
Finite Element Software ◽  
Time Stepping ◽  
Backward Euler ◽  
Shaped Domain

Although two-dimensional (2D) parabolic integro-differential equations (PIDEs) arise in many physical contexts, there is no generally available software that is able to solve them numerically. To remedy this situation, in this article, we provide a compact implementation for solving 2D PIDEs using the finite element method (FEM) on unstructured grids. Piecewise linear finite element spaces on triangles are used for the space discretization, whereas the time discretization is based on the backward-Euler and the Crank–Nicolson methods. The quadrature rules for discretizing the Volterra integral term are chosen so as to be consistent with the time-stepping schemes; a more efficient version of the implementation that uses a vectorization technique in the assembly process is also presented. The compactness of the approach is demonstrated using the software Matrix Laboratory (MATLAB). The efficiency is demonstrated via a numerical example on an L-shaped domain, for which a comparison is possible against the commercially available finite element software COMSOL Multiphysics. Moreover, further consideration indicates that COMSOL Multiphysics cannot be directly applied to 2D PIDEs containing more complex kernels in the Volterra integral term, whereas our method can. Consequently, the subroutines we present constitute a valuable open and validated resource for solving more general 2D PIDEs.

scholarly journals A Finite Element Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 717-725 ◽  
Author(s):  
Vidar Thomée
Keyword(s):  
Finite Element ◽  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Euler Approximation ◽  
Convection Diffusion ◽  
Time Stepping ◽  
Backward Euler ◽  
Spatially Periodic ◽  
Forward Euler

AbstractFor a spatially periodic convection-diffusion problem, we analyze a time stepping method based on Lie splitting of a spatially semidiscrete finite element solution on time steps of length k, using the backward Euler method for the diffusion part and a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {k/m} for the convection part. This complements earlier work on time splitting of the problem in a finite difference context.

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.

SIMULATION STUDY ON NON-HOMOGENOUS SYSTEM OF NON-INVASIVE ERT USING COMSOL MULTIPHYSICS

2015 ◽  
Vol 77 (17) ◽  
Author(s):  
Yasmin Abdul Wahab ◽  
Ruzairi Abdul Rahim ◽  
Mohd Hafiz Fazalul Rahiman ◽  
Leow Pei Ling ◽  
Suzzana Ridzuan Aw ◽  
...  
Keyword(s):  
Finite Element ◽  
Simulation Study ◽  
Direct Contact ◽  
Comsol Multiphysics ◽  
Finite Element Software ◽  
Non Invasive ◽  
System A ◽  
Process Tomography ◽  
Gas Medium ◽  
Air Medium

The non-invasive sensing technique is one of the favourite sensing techniques applied in the process tomography because it has not a direct contact with the medium of interest. The objective of this paper is to analyse the simulation of the non-homogenous system of the non-invasive ERT using finite element software; COMSOL Multiphysics. In this simulation, the liquid-air medium is chosen as the non-homogenous system. A different analysis of the non-homogenous system in term of the different position of the single air, different size of the single air and the multiple air inside the vessel were investigated in this paper. As a result, the location, size and multiple air inside the pipe will influence the output of the non-invasive ERT system. A liquid-gas medium of non-homogenous ERT system will have a good response if the air is located near the source, the size of the air is large enough and it has multiple air locations inside the pipe.

scholarly journals Second-order time discretization with finite-element method for partial integro-differential equations with a weakly singular kernel

1999 ◽  
Vol 40 (4) ◽  
pp. 513-524
Author(s):  
Chang Ho Kim ◽  
U Jin Choi
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Time Discretization ◽  
Second Order ◽  
Singular Kernel ◽  
Element Approximation ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Element Method

AbstractWe propose the second-order time discretization scheme with the finite-element approximation for the partial integro-differential equations with a weakly singular kernel. The space discretization is based on the finite element method and the time discretization is based on the Crank-Nicolson scheme with a graded mesh. We show the stability of the scheme and obtain the second-order convergence result for the fully discretized scheme.

scholarly journals On Positivity and Maximum-Norm Contractivity in Time Stepping Methods for Parabolic Equations

2010 ◽  
Vol 10 (4) ◽  
pp. 421-443 ◽  
Author(s):  
A.H. Schatz ◽  
V. Thomèe ◽  
L.B. Wahlbin
Keyword(s):  
Piecewise Linear ◽  
Small Time ◽  
Maximum Norm ◽  
Lumped Mass ◽  
Finite Element Approximations ◽  
Time Stepping ◽  
Backward Euler ◽  
Discrete Analogues ◽  
Carry Over ◽  
Standard Galerkin Method

Abstract In an earlier paper the last two authors studied spatially semidiscrete piecewise linear finite element approximations of the heat equation and showed that, in the case of the standard Galerkin method, the solution operator of the initial-value problem is neither positive nor contractive in the maximum-norm for small time, but that for the lumped mass method these properties hold, if the triangulations are essentially of Delaunay type. In this paper we continue the study by considering fully discrete analogues obtained by discretization also in time. The above properties then carry over to the backward Euler time stepping method, but for other methods the results are more restrictive. We discuss in particular the θ-method and the (0; 2) Padé approximation in one space dimension.

Crouzeix–Raviart Finite Element Approximation for the Parabolic Obstacle Problem

2020 ◽  
Vol 20 (2) ◽  
pp. 273-292 ◽  
Author(s):  
Thirupathi Gudi ◽  
Papri Majumder
Keyword(s):  
Finite Element ◽  
Obstacle Problem ◽  
Time Discretization ◽  
Finite Element Approximation ◽  
Optimal Order ◽  
Element Approximation ◽  
Fully Discrete ◽  
Backward Euler ◽  
Space Discretization ◽  
Speed Of Propagation

AbstractWe introduce and study a fully discrete nonconforming finite element approximation for a parabolic variational inequality associated with a general obstacle problem. The method comprises of the Crouzeix–Raviart finite element method for space discretization and implicit backward Euler scheme for time discretization. We derive an error estimate of optimal order {\mathcal{O}(h+\Delta t)} in a certain energy norm defined precisely in the article. We only assume the realistic regularity {u_{t}\in L^{2}(0,T;L^{2}(\Omega))} and moreover the analysis is performed without any assumptions on the speed of propagation of the free boundary. We present a numerical experiment to illustrate the theoretical order of convergence derived in the article.

scholarly journals Strength optimization of structural elements by means of optimal control

2019 ◽  
Vol 262 ◽  
pp. 10006
Author(s):  
Dorota Jasinska ◽  
Leszek Mikulski
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Differential Equations ◽  
Control Theory ◽  
Boundary Value Problem ◽  
Structural Elements ◽  
Finite Element Software ◽  
Portal Frame ◽  
Multipoint Boundary Value Problem ◽  
The Web

This paper investigates the optimal shaping of the web height of an I-section steel portal frame. The problem is formulated as a control theory task. From a mathematical perspective, the task involves solving the multipoint boundary value problem for the system of forty-three differential equations. The solution is compared to results obtained from the finite element software Abaqus.

scholarly journals Error Estimates of a Continuous Galerkin Time Stepping Method for Subdiffusion Problem

2021 ◽  
Vol 88 (3) ◽  
Author(s):  
Yuyuan Yan ◽  
Bernard A. Egwu ◽  
Zongqi Liang ◽  
Yubin Yan
Keyword(s):  
Piecewise Linear ◽  
Time Discretization ◽  
Convergence Order ◽  
Step Size ◽  
Time Step ◽  
Transform Method ◽  
Nonsmooth Data ◽  
Time Step Size ◽  
Time Stepping ◽  
Continuous Galerkin

AbstractA continuous Galerkin time stepping method is introduced and analyzed for subdiffusion problem in an abstract setting. The approximate solution will be sought as a continuous piecewise linear function in time t and the test space is based on the discontinuous piecewise constant functions. We prove that the proposed time stepping method has the convergence order $$O(\tau ^{1+ \alpha }), \, \alpha \in (0, 1)$$ O ( τ 1 + α ) , α ∈ ( 0 , 1 ) for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where $$\tau $$ τ is the time step size. This convergence order is higher than the convergence orders of the popular convolution quadrature methods (e.g., Lubich’s convolution methods) and L-type methods (e.g., L1 method), which have only $$O(\tau )$$ O ( τ ) convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity.

scholarly journals Fireshape: a shape optimization toolbox for Firedrake

2021 ◽  
Vol 63 (5) ◽  
pp. 2553-2569
Author(s):  
Alberto Paganini ◽  
Florian Wechsung
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Shape Optimization ◽  
Open Source ◽  
Optimization Problems ◽  
Moving Mesh ◽  
Moving Mesh Method ◽  
Finite Element Software ◽  
Mesh Method

AbstractWe introduce Fireshape, an open-source and automated shape optimization toolbox for the finite element software Firedrake. Fireshape is based on the moving mesh method and allows users with minimal shape optimization knowledge to tackle with ease challenging shape optimization problems constrained to partial differential equations (PDEs).

Flexural Wave Propagation Characteristics of Lattice Sandwich Plates

2013 ◽  
Vol 753-755 ◽  
pp. 857-860 ◽  
Author(s):  
Shu Lang Tao ◽  
Gui Lan Yu ◽  
Zong Jian Yao
Keyword(s):  
Finite Element ◽  
Wave Propagation ◽  
Band Gaps ◽  
Comsol Multiphysics ◽  
Flexural Wave ◽  
Sandwich Plates ◽  
Propagation Characteristics ◽  
Finite Element Software ◽  
Flexural Wave Propagation ◽  
Lattice Pattern

This paper is aimed to study flexural wave propagation characteristics of lattice sandwich plates. Based on Blochs theorem, band structure of flexural wave propagation in the plate is obtained by commercial finite element software Comsol Multiphysics. Meanwhile, frequency response is obtained and its maximum attenuation is exactly corresponding to the band gaps. Finally, effects of lattice pattern on band gaps are introduced.

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