On the Construction and Analysis of High Order Locally Conservative Finite Volume-Type Methods for One-Dimensional Elliptic Problems

Plexousakis, Michael; Zouraris, Georgios E.

doi:10.1137/s0036142902406302

Cited by 58 publications

(40 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The analysis in [16] shows that choosing carefully the control volumes the method achieves optimal order of convergence in the H 1 , L 2 and L ∞ norm, i.e. is the same with the corresponding order of convergence of the finite element method.…”

Section: Numerical Solution Of a Non-local Elliptic Problem 769mentioning

confidence: 93%

“…The methods (8) and (10) generalize a method developed in Proposition 3.7 of [16] for a general linear two-point boundary value problem with Dirichlet-Neumann boundary conditions. The value of ρ is important because it affects the order of convergence of the finite volume method.…”

Section: Formulation Of the Numerical Methodsmentioning

confidence: 99%

“…The value of ρ is important because it affects the order of convergence of the finite volume method. The analysis of [16] shows that in the linear case the finite volume method attains optimal order of convergence in the H 1 norm for ρ ∈ (0, 1), and in addition optimal order of convergence in the L 2 and L ∞ norm for the special values ρ = 1 2 ± √ 3 6 . We expect that the order of convergence of the methods (FVR) and (FVD) depends on ρ in an analogous manner, and that is confirmed numerically computing the rate of convergence in a case with known solution (see Sections 3.1 and 3.2).…”

Section: Formulation Of the Numerical Methodsmentioning

confidence: 99%

“…The outcome of both method is a function, which is continuous on D and piecewise polynomial of degree less than or equal to 2 over a given partition of D. The formulation of the finite element method is standard and is based on a variational formulation of the continuous problem. The finite volume method is new and is a generalization of a method proposed and analyzed in [16] for a general linear two-point boundary value problem with Dirichlet-Neumann boundary conditions. The reason that the finite volume method is used in the numerical approximation of heat flow problems (see, e.g., [8]) is the fact that the approximation w FV that derives is locally conservative, i.e.…”

Section: Numerical Solution Of a Non-local Elliptic Problem 769mentioning

confidence: 99%

See 3 more Smart Citations

Numerical Solution of a Non-Local Elliptic Problem Modeling a Thermistor with a Finite Element and a Finite Volume Method

Nikolopoulos

Zouraris

2008

Progress in Industrial Mathematics at ECMI 2006

View full text Add to dashboard Cite

Abstract. We consider the following non-local elliptic boundary value problem:where α and λ are positive constants and f is a function satisfyingThe solution of the equation represents the steady state of a thermistor device. The problem has a unique solution for a critical value λ * of the parameter λ, at least two solutions for λ < λ * and has no solution for λ > λ * . We apply a finite element and a finite volume method in order to find a numerical approximation of the solution of the problem from the space of continuous piecewise quadratic functions, for the case that λ < λ * and for the stable branch of the bifurcation diagram. A comparison of these two methods is made regarding their order of convergence for f (s) = e −s and f (s) = (1 + s) −2 . Also, for the same equation but with Dirichlet boundary conditions, a situation where the solution is unique for λ < λ * , a similar comparison of the finite element and the finite volume method is presented.

show abstract

Section: Numerical Solution Of a Non-local Elliptic Problem 769mentioning

confidence: 93%

Section: Formulation Of the Numerical Methodsmentioning

confidence: 99%

Section: Formulation Of the Numerical Methodsmentioning

confidence: 99%

Section: Numerical Solution Of a Non-local Elliptic Problem 769mentioning

confidence: 99%

See 2 more Smart Citations

Numerical Solution of a Non-Local Elliptic Problem Modeling a Thermistor with a Finite Element and a Finite Volume Method

Nikolopoulos

Zouraris

2008

Progress in Industrial Mathematics at ECMI 2006

View full text Add to dashboard Cite

show abstract

“…Let us finally mention some related results in the context of vertex-centered finite volume (element) methods [6,9,19,23,24,25].…”

mentioning

confidence: 99%

A Posteriori Error Estimation for the Discrete Duality Finite Volume Discretization of the Laplace Equation

Penel

Rosenbaum

2009

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Abstract. An efficient and fully computable a posteriori error bound is derived for the discrete duality finite volume discretization of the Laplace equation on very general twodimensional meshes. The main ingredients are the equivalence of this method with a finite element like scheme and tools from the finite element framework. Numerical tests are performed with a stiff solution on highly nonconforming locally refined meshes and with a singular solution on triangular meshes.

show abstract

A posteriori error analysis of nonconforming finite volume elements for general second‐order elliptic PDEs

Yang

2011

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

On the Construction and Analysis of High Order Locally Conservative Finite Volume-Type Methods for One-Dimensional Elliptic Problems

Cited by 58 publications

References 33 publications

Numerical Solution of a Non-Local Elliptic Problem Modeling a Thermistor with a Finite Element and a Finite Volume Method

Numerical Solution of a Non-Local Elliptic Problem Modeling a Thermistor with a Finite Element and a Finite Volume Method

A Posteriori Error Estimation for the Discrete Duality Finite Volume Discretization of the Laplace Equation

A posteriori error analysis of nonconforming finite volume elements for general second‐order elliptic PDEs

Contact Info

Product

Resources

About