Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains

Hengguang Li, Anna L. Mazzucato, Victor Nistor

Research output: Contribution to journalArticle

34 Citations (Scopus)

Abstract

We study theoretical and practical issues arising in the implementation of the Finite Element Method for a strongly elliptic second order equation with jump discontinuities in its coefficients on a polygonal domain λ that may have cracks or vertices that touch the boundary. We consider in particular the equation-div(Aδu) = f ΕHm-1λ with mixed boundary conditions, where the matrix A has variable, piecewise smooth coefficients. We establish regularity and Fredholm results and, under some additional conditions, we also establish well-posedness in weighted Sobolev spaces. When Neumann boundary conditions are imposed on adjacent sides of the polygonal domain, we obtain the decomposition u = ureg + φ, into a function u reg with better decay at the vertices and a function that is locally constant near the vertices, thus proving well-posedness in an augmented space. The theoretical analysis yields interpolation estimates that are then used to construct improved graded meshes recovering the (quasi-)optimal rate of convergence for piecewise polynomials of degree m ≥ 1. Several numerical tests are included.

Original languageEnglish (US)
Pages (from-to)41-69
Number of pages29
JournalElectronic Transactions on Numerical Analysis
Volume37
StatePublished - Dec 1 2010

Fingerprint

Mixed Boundary Value Problem
Well-posedness
Finite Element Method
Graded Meshes
Second Order Elliptic Equations
Optimal Rate of Convergence
Weighted Sobolev Spaces
Mixed Boundary Conditions
Piecewise Polynomials
Coefficient
Neumann Boundary Conditions
Discontinuity
Theoretical Analysis
Crack
Jump
Adjacent
Interpolate
Regularity
Decay
Decompose

All Science Journal Classification (ASJC) codes

  • Analysis

Cite this

@article{9faab2a19e78431fa3e9df1b1ce2e5a9,
title = "Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains",
abstract = "We study theoretical and practical issues arising in the implementation of the Finite Element Method for a strongly elliptic second order equation with jump discontinuities in its coefficients on a polygonal domain λ that may have cracks or vertices that touch the boundary. We consider in particular the equation-div(Aδu) = f ΕHm-1λ with mixed boundary conditions, where the matrix A has variable, piecewise smooth coefficients. We establish regularity and Fredholm results and, under some additional conditions, we also establish well-posedness in weighted Sobolev spaces. When Neumann boundary conditions are imposed on adjacent sides of the polygonal domain, we obtain the decomposition u = ureg + φ, into a function u reg with better decay at the vertices and a function that is locally constant near the vertices, thus proving well-posedness in an augmented space. The theoretical analysis yields interpolation estimates that are then used to construct improved graded meshes recovering the (quasi-)optimal rate of convergence for piecewise polynomials of degree m ≥ 1. Several numerical tests are included.",
author = "Hengguang Li and Mazzucato, {Anna L.} and Victor Nistor",
year = "2010",
month = "12",
day = "1",
language = "English (US)",
volume = "37",
pages = "41--69",
journal = "Electronic Transactions on Numerical Analysis",
issn = "1068-9613",
publisher = "Kent State University",

}

Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains. / Li, Hengguang; Mazzucato, Anna L.; Nistor, Victor.

In: Electronic Transactions on Numerical Analysis, Vol. 37, 01.12.2010, p. 41-69.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains

AU - Li, Hengguang

AU - Mazzucato, Anna L.

AU - Nistor, Victor

PY - 2010/12/1

Y1 - 2010/12/1

N2 - We study theoretical and practical issues arising in the implementation of the Finite Element Method for a strongly elliptic second order equation with jump discontinuities in its coefficients on a polygonal domain λ that may have cracks or vertices that touch the boundary. We consider in particular the equation-div(Aδu) = f ΕHm-1λ with mixed boundary conditions, where the matrix A has variable, piecewise smooth coefficients. We establish regularity and Fredholm results and, under some additional conditions, we also establish well-posedness in weighted Sobolev spaces. When Neumann boundary conditions are imposed on adjacent sides of the polygonal domain, we obtain the decomposition u = ureg + φ, into a function u reg with better decay at the vertices and a function that is locally constant near the vertices, thus proving well-posedness in an augmented space. The theoretical analysis yields interpolation estimates that are then used to construct improved graded meshes recovering the (quasi-)optimal rate of convergence for piecewise polynomials of degree m ≥ 1. Several numerical tests are included.

AB - We study theoretical and practical issues arising in the implementation of the Finite Element Method for a strongly elliptic second order equation with jump discontinuities in its coefficients on a polygonal domain λ that may have cracks or vertices that touch the boundary. We consider in particular the equation-div(Aδu) = f ΕHm-1λ with mixed boundary conditions, where the matrix A has variable, piecewise smooth coefficients. We establish regularity and Fredholm results and, under some additional conditions, we also establish well-posedness in weighted Sobolev spaces. When Neumann boundary conditions are imposed on adjacent sides of the polygonal domain, we obtain the decomposition u = ureg + φ, into a function u reg with better decay at the vertices and a function that is locally constant near the vertices, thus proving well-posedness in an augmented space. The theoretical analysis yields interpolation estimates that are then used to construct improved graded meshes recovering the (quasi-)optimal rate of convergence for piecewise polynomials of degree m ≥ 1. Several numerical tests are included.

UR - http://www.scopus.com/inward/record.url?scp=78651482860&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78651482860&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:78651482860

VL - 37

SP - 41

EP - 69

JO - Electronic Transactions on Numerical Analysis

JF - Electronic Transactions on Numerical Analysis

SN - 1068-9613

ER -