Métodos miméticos de pasos fraccionarios - E-Archivo - Universidad ...

Contents
1 Introduction 1
2 Cell-node mimetic domain decomposition methods 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Properties of the continuous operators . . . . . . . . . . . . . . . 11
2.3 Spatial semidiscretization: a cell-node MFD method . . . . . . . 12
2.3.1 Spaces of semidiscrete functions . . . . . . . . . . . . . . . 13
2.3.2 Discrete divergence operator . . . . . . . . . . . . . . . . 14
2.3.3 Discrete inner products . . . . . . . . . . . . . . . . . . . 16
2.3.4 Discrete gradient operator . . . . . . . . . . . . . . . . . . 17
2.3.5 The MFD semidiscrete scheme . . . . . . . . . . . . . . . 19
2.3.6 Discrete operators on a rectangular grid . . . . . . . . . . 20
2.4 Time integration: a linearly implicit FSRKm+1 method . . . . . 23
2.4.1 Domain decomposition operator splitting . . . . . . . . . 23
2.4.2 The linearly implicit FSRKm+1 method . . . . . . . . . . 26
2.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.1 A problem on a pseudo-random grid . . . . . . . . . . . . 29
2.5.2 A problem on a smooth grid . . . . . . . . . . . . . . . . 34
3 Cell-edge mimetic domain decomposition methods 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Properties of the continuous operators . . . . . . . . . . . . . . . 41
3.3 Spatial semidiscretization: a cell-edge MFD method . . . . . . . 44
3.3.1 Spaces of semidiscrete functions . . . . . . . . . . . . . . . 44
3.3.2 Discrete extended divergence operator . . . . . . . . . . . 46
3.3.3 Discrete inner products . . . . . . . . . . . . . . . . . . . 47
3.3.4 Discrete flux operator . . . . . . . . . . . . . . . . . . . . 51
3.3.5 The MFD semidiscrete scheme . . . . . . . . . . . . . . . 55
3.3.6 Discrete operators on a rectangular grid . . . . . . . . . . 57
3.4 The mixed finite element method . . . . . . . . . . . . . . . . . . 59
3.4.1 Preliminaries: the MFD method revisited . . . . . . . . . 59
3.4.2 The weak formulation . . . . . . . . . . . . . . . . . . . . 62