### 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 ΕH^{m-1}λ 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 = u_{reg} + 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 language | English (US) |
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Pages (from-to) | 41-69 |

Number of pages | 29 |

Journal | Electronic Transactions on Numerical Analysis |

Volume | 37 |

State | Published - Dec 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Analysis

### Cite this

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*Electronic Transactions on Numerical Analysis*, vol. 37, pp. 41-69.

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

Research output: Contribution to journal › Article

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

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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 -