## Finite difference methods on irregular networks: a generalized approach to second order elliptic problems |

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### Contents

INTRODUCTION | 9 |

BOUNDARY VALUE PROBLEMS AND IRREGULAR NETWORKS | 17 |

CONSTRUCTION OF FINITE DIFFERENCE APPROXIMATIONS | 40 |

Copyright | |

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affine transformation angles Appendix EX approximation error asymptotic behaviour balance equations boundary conditions bounds boxes Bramble-Hilbert Lemma C-norm coefficients considered convergence convergent boundary cubature defined denote depend derived difference operators difference quotients discrete analogues domain elliptic error estimates error functionals estimation regions FDSs A.y fh(x finite element fulfilled Furthermore Green's formula grid functions grid points grid regularity condition HEINRICH inequalities kh(x KVu,n)ds line integral linear locally irregular networks matrix MD-schemes mesh segment method MD method PB midpoint mixed boundary conditions monograph Moreover networks of triangles norms PB-schemes perpendicular bisectors polygonal positive definiteness priori estimates Proof proved rectangles Remark respect to h rhombus right-hand side satisfied scalar product secondary networks Section seminorms smoothness assumptions Sobolev spaces solution subsets sufficiently small symmetry take into account taken Taylor's expansion Theorem 3.1 tion trace theorems transformation upwind triangles vector xeco