scholarly journals Numerical method for solving the piecewise constant source inverse problem of an elliptic equation from a partial boundary observation data

2021 ◽  
Vol 2092 (1) ◽  
pp. 012006
Author(s):  
D Kh Ivanov ◽  
A E Kolesov ◽  
P N Vabishchevich
Keyword(s):  
Inverse Problem ◽  
Elliptic Equation ◽  
Source Term ◽  
Numerical Solutions ◽  
Finite Element Approximation ◽  
Normal Derivative ◽  
Test Problems ◽  
Element Approximation ◽  
Observation Data ◽  
Additional Information

Abstract We present results of numerical investigation of the source term recovery in a boundary value problem for an elliptic equation. An additional information about the solution is considered as its normal derivative taken on a part of the boundary. Such source inverse problem is related with inverse gravimetry problem of determining an inhomogeneity from gravitational potential anomalies on the Earth’s surface. We propose an iterative method for numerical recovery of the source term on the base of minimization of the observation residual by a gradient type method. The numerical implementation is based on finite element approximation using the FEniCS scientific computing platform and the dolfin-adjoint package. The capabilities of the developed computational algorithm are illustrated by results of numerical solutions of two dimensional test problems.

Numerical Simulation for Interaction Problem of Largely Deformed Membrane and Fluid

2006 ◽  
Author(s):  
Takahiro Yamada ◽  
Tomohiro Kayane ◽  
Yoshiaki Itoh ◽  
Ryuko Ootsuka
Keyword(s):  
Numerical Solutions ◽  
Numerical Procedure ◽  
Viscous Flows ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Conservation Of Energy ◽  
Fluid Structure ◽  
Seidel Method ◽  
Coupling Algorithm ◽  
Interaction Problem

In this paper, a numerical procedure for interaction problem of largely deformed membrane and fluid is proposed. Numerical computations of deformable structures in viscous flows often encounter numerical instability, which arises from incompatibility of boundary conditions on a fluid-structure interface. In this work, two aspects of compatibility condition including the balance of energy on the fluid-structure interface and the coupling algorithm are discussed and measures to improve attainment of them in numerical procedures are proposed. In this work, a finite difference method with overlapped meshes is employed to solve a fluid problem accurately. For a membrane, a finite element approximation with the energy-momentum method for the temporal discretization is applied to ensure the conservation of energy and unconditional numerical stability. Numerical solutions of these two subsystems are coupled by the block Gauss-Seidel method. Numerical results are given to show numerical properties including stability of the proposed procedure.

A refined convergence analysis of multigrid algorithms for elliptic equations

2015 ◽  
Vol 13 (03) ◽  
pp. 255-290 ◽  
Author(s):  
Qianshun Chang ◽  
Rong-Qing Jia
Keyword(s):  
General Theory ◽  
Convergence Analysis ◽  
Elliptic Equations ◽  
Numerical Solutions ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Multigrid Algorithm ◽  
Seidel Method ◽  
Wide Range ◽  
Additional Regularity

Multigrid algorithms, in particular, multigrid V-cycles, are investigated in this paper. We establish a general theory for convergence of the multigrid algorithm under certain approximation conditions and smoothing conditions. Our smoothing conditions are satisfied by commonly used smoothing operators including the standard Gauss–Seidel method. Our approximation conditions are verified for finite element approximation to numerical solutions of elliptic partial differential equations without any requirement of additional regularity of the solution. Our convergence analysis of multigrid algorithms can be applied to a wide range of problems. Numerical examples are also provided to illustrate the general theory.

scholarly journals Superconvergence of the function value for pentahedral finite elements for an elliptic equation with varying coefficients

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jinghong Liu ◽  
Qiding Zhu
Keyword(s):  
Finite Element ◽  
Green Function ◽  
Finite Elements ◽  
Elliptic Equation ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
True Solution ◽  
Varying Coefficients ◽  
Fundamental Estimate

AbstractIn this article, for an elliptic equation with varying coefficients, we first derive an interpolation fundamental estimate for the $\mathcal{P}_{2}(x,y)\otimes \mathcal{P}_{2}(z)$P2(x,y)⊗P2(z) pentahedral finite element over uniform partitions of the domain. Then combined with the estimate for the $W^{2,1}$W2,1-seminorm of the discrete Green function, superconvergence of the function value between the finite element approximation and the corresponding interpolant to the true solution is given.

scholarly journals A Mixed Finite Element Formulation for the Conservative Fractional Diffusion Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Suxiang Yang ◽  
Huanzhen Chen
Keyword(s):  
Finite Element ◽  
Elliptic Equation ◽  
Error Estimates ◽  
Singular Term ◽  
Mixed Finite Element ◽  
Finite Element Approximation ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Element Formulation ◽  
Optimal Error

We consider a boundary-value problem of one-side conservative elliptic equation involving Riemann-Liouville fractional integral. The appearance of the singular term in the solution leads to lower regularity of the solution of the equation, so to the lower order convergence rate for the numerical solution. In this paper, by the dividing of equation, we drop the lower regularity term in the solution successfully and get a new fractional elliptic equation which has full regularity. We present a theoretical framework of mixed finite element approximation to the new fractional elliptic equation and derive the error estimates for unknown function, its derivative, and fractional-order flux. Some numerical results are illustrated to confirm the optimal error estimates.

scholarly journals On the numerical solution of nonlinear eigenvalue problems for the Monge-Ampère operator

2020 ◽  
Vol 26 ◽  
pp. 118
Author(s):  
Roland Glowinski ◽  
Shingyu Leung ◽  
Hao Liu ◽  
Jianliang Qian
Keyword(s):  
Exact Solutions ◽  
Numerical Solution ◽  
Eigenvalue Problems ◽  
Gradient Flow ◽  
Finite Element Approximation ◽  
Test Problems ◽  
Element Approximation ◽  
Nonlinear Eigenvalue Problems ◽  
Monge Ampere ◽  
Nonlinear Eigenvalue

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Ampère operator v → det D2v. The methodology we employ relies on the following ingredients: (i) a divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step h → 0. We considered also test problems with no known exact solutions.

Well-posedness of determining the source term of an elliptic equation

1994 ◽  
Vol 50 (3) ◽  
pp. 383-398 ◽  
Author(s):  
Wenhuan Yu
Keyword(s):  
Inverse Problem ◽  
Elliptic Equation ◽  
Banach Spaces ◽  
Continuous Dependence ◽  
Source Term ◽  
Well Posedness ◽  
Elimination Method

In this paper the inverse problem for determining the source term of a linear, uniformly elliptic equation is investigated. The uniqueness of the inverse problem is proved under mild assumptions by use of the orthogonality method and an elimination method. The existence of the inverse problem is proved by means of the theory of solvable operators between Banach spaces, moreover, the continuous dependence of the solution to the inverse problem on measurement is also obtained.

scholarly journals Numerical analysis of the mixed finite element method for the neutron diffusion eigenproblem with heterogeneous coefficients

2018 ◽  
Vol 52 (5) ◽  
pp. 2003-2035
Author(s):  
P. Ciarlet ◽  
L. Giret ◽  
E. Jamelot ◽  
F.D. Kpadonou
Keyword(s):  
Finite Element ◽  
Domain Decomposition ◽  
Decomposition Method ◽  
Source Term ◽  
Domain Decomposition Method ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Different Characteristics

We study first the convergence of the finite element approximation of the mixed diffusion equations with a source term, in the case where the solution is of low regularity. Such a situation commonly arises in the presence of three or more intersecting material components with different characteristics. Then we focus on the approximation of the associated eigenvalue problem. We prove spectral correctness for this problem in the mixed setting. These studies are carried out without, and then with a domain decomposition method. The domain decomposition method can be non-matching in the sense that the traces of the finite element spaces may not fit at the interface between subdomains. Finally, numerical experiments illustrate the accuracy of the method.

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