### Abstract

We consider the model problem -Δu + V u = f ∈ Ω with suitable boundary conditions on ∂Ω where Ω is a bounded polyhedral domain in ℝ^{d}, d = 2; 3, and V is a possibly singular potential. We study efficient finite element discretizations of our problem following Numer. Funct. Anal. Optim., 28(7-8):775-824, 2007, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 55(103):157-178, 2012, and a few other more recent papers. Under some additonal mild assumptions, we show how to construct sequences of meshes such that the sequence of Galerkin approximations u _{κ} ∈ S_{κ} achieve h^{m}-quasi- optimal rates of convergence in the sense that ∥u - u_{κ} ∥H^{1}(Ω) ≤ C dim(S_{κ}) ^{m/d}∥f∥H^{m-1}(Ω). Our meshes defining the Finite Element spaces Sk are suitably graded towards the singularities. They are topologically equivalent to the meshes obtained by uniform refinement and hence their construction is easy to implement. We explain in detail the mesh refinement for four typical problems: for mixed boundary value/transmission problems for which V = 0, for Schrödinger type operators with an inverse square potential V on a polygonal domain in 2D, for Schrödinger type operators with a periodic inverse square potential V in 3D, and for the Poisson problem on a three dimensional domain. These problems are listed in the increasing order of complexity. The transmission problems considered here include the cases of multiple junction points.

Original language | English (US) |
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Title of host publication | ECCOMAS 2012 - European Congress on Computational Methods in Applied Sciences and Engineering, e-Book Full Papers |

Pages | 9003-9014 |

Number of pages | 12 |

State | Published - 2012 |

Event | 6th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2012 - Vienna, Austria Duration: Sep 10 2012 → Sep 14 2012 |

### Other

Other | 6th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2012 |
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Country | Austria |

City | Vienna |

Period | 9/10/12 → 9/14/12 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*ECCOMAS 2012 - European Congress on Computational Methods in Applied Sciences and Engineering, e-Book Full Papers*(pp. 9003-9014)

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*ECCOMAS 2012 - European Congress on Computational Methods in Applied Sciences and Engineering, e-Book Full Papers.*pp. 9003-9014, 6th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2012, Vienna, Austria, 9/10/12.

**Anisotropic graded meshes and quasi-optimal rates of convergence for the FEM on polyhedral domains in 3D.** / Bǎcuţǎ, Constantin; Li, Hengguang; Nistor, Victor.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Anisotropic graded meshes and quasi-optimal rates of convergence for the FEM on polyhedral domains in 3D

AU - Bǎcuţǎ, Constantin

AU - Li, Hengguang

AU - Nistor, Victor

PY - 2012

Y1 - 2012

N2 - We consider the model problem -Δu + V u = f ∈ Ω with suitable boundary conditions on ∂Ω where Ω is a bounded polyhedral domain in ℝd, d = 2; 3, and V is a possibly singular potential. We study efficient finite element discretizations of our problem following Numer. Funct. Anal. Optim., 28(7-8):775-824, 2007, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 55(103):157-178, 2012, and a few other more recent papers. Under some additonal mild assumptions, we show how to construct sequences of meshes such that the sequence of Galerkin approximations u κ ∈ Sκ achieve hm-quasi- optimal rates of convergence in the sense that ∥u - uκ ∥H1(Ω) ≤ C dim(Sκ) m/d∥f∥Hm-1(Ω). Our meshes defining the Finite Element spaces Sk are suitably graded towards the singularities. They are topologically equivalent to the meshes obtained by uniform refinement and hence their construction is easy to implement. We explain in detail the mesh refinement for four typical problems: for mixed boundary value/transmission problems for which V = 0, for Schrödinger type operators with an inverse square potential V on a polygonal domain in 2D, for Schrödinger type operators with a periodic inverse square potential V in 3D, and for the Poisson problem on a three dimensional domain. These problems are listed in the increasing order of complexity. The transmission problems considered here include the cases of multiple junction points.

AB - We consider the model problem -Δu + V u = f ∈ Ω with suitable boundary conditions on ∂Ω where Ω is a bounded polyhedral domain in ℝd, d = 2; 3, and V is a possibly singular potential. We study efficient finite element discretizations of our problem following Numer. Funct. Anal. Optim., 28(7-8):775-824, 2007, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 55(103):157-178, 2012, and a few other more recent papers. Under some additonal mild assumptions, we show how to construct sequences of meshes such that the sequence of Galerkin approximations u κ ∈ Sκ achieve hm-quasi- optimal rates of convergence in the sense that ∥u - uκ ∥H1(Ω) ≤ C dim(Sκ) m/d∥f∥Hm-1(Ω). Our meshes defining the Finite Element spaces Sk are suitably graded towards the singularities. They are topologically equivalent to the meshes obtained by uniform refinement and hence their construction is easy to implement. We explain in detail the mesh refinement for four typical problems: for mixed boundary value/transmission problems for which V = 0, for Schrödinger type operators with an inverse square potential V on a polygonal domain in 2D, for Schrödinger type operators with a periodic inverse square potential V in 3D, and for the Poisson problem on a three dimensional domain. These problems are listed in the increasing order of complexity. The transmission problems considered here include the cases of multiple junction points.

UR - http://www.scopus.com/inward/record.url?scp=84871632193&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871632193&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9783950353709

SP - 9003

EP - 9014

BT - ECCOMAS 2012 - European Congress on Computational Methods in Applied Sciences and Engineering, e-Book Full Papers

ER -