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x contents 3.4.3 Finite element mappings . . . . . . . . . . . . . . . . . . . 63 3.4.4 MFE spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4.5 Relating MFD and MFE methods . . . . . . . . . . . . . 66 3.5 Convergence analysis of the semidiscrete scheme . . . . . . . . . 71 3.5.1 The L 2 (Ω) and Raviart–Thomas projections . . . . . . . . 71 3.5.2 The elliptic MFE projection . . . . . . . . . . . . . . . . . 72 3.5.3 Convergence of the MFD method . . . . . . . . . . . . . . 74 3.6 Time integration: a linearly implicit FSRKm+1 method . . . . . 77 3.6.1 Domain decomposition operator splitting . . . . . . . . . 77 3.6.2 The linearly implicit FSRKm+1 method . . . . . . . . . . 78 3.7 Convergence analysis of the totally discrete scheme . . . . . . . . 80 3.7.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.7.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.7.3 Convergence of the MFD-FSRKm+1 method . . . . . . . 88 3.8 The mimetic fractional step method on non-orthogonal grids . . 90 3.8.1 Time integration . . . . . . . . . . . . . . . . . . . . . . . 90 3.8.2 Spatial discretization . . . . . . . . . . . . . . . . . . . . . 91 3.8.3 Solving a system of equations with non-local stencil . . . 94 3.9 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . 95 3.9.1 A problem on a rectangular grid . . . . . . . . . . . . . . 95 3.9.2 A problem on a non-orthogonal grid . . . . . . . . . . . . 97 4 Cell-centered finite difference alternating direction implicit methods 103 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2 Spatial semidiscretization: a CCFD method . . . . . . . . . . . . 106 4.2.1 The expanded weak formulation . . . . . . . . . . . . . . 107 4.2.2 Finite element mappings . . . . . . . . . . . . . . . . . . . 108 4.2.3 Expanded MFE spaces . . . . . . . . . . . . . . . . . . . . 112 4.2.4 Towards a finite difference scheme: a quadrature rule . . . 113 4.2.5 The CCFD method . . . . . . . . . . . . . . . . . . . . . 115 4.3 Convergence analysis of the semidiscrete scheme . . . . . . . . . 121 4.3.1 The L 2 (Ω) and Raviart–Thomas projections . . . . . . . . 121 4.3.2 The expanded elliptic MFE projection . . . . . . . . . . . 122 4.3.3 Convergence of the CCFD method . . . . . . . . . . . . . 124 4.4 Time integration: a linearly implicit FSRKm+1 method . . . . . 126 4.5 Convergence analysis of the totally discrete scheme . . . . . . . . 128 4.5.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.5.2 Stability: the von Neumann analysis . . . . . . . . . . . . 130 4.5.3 Convergence of the CCFD-FSRKm+1 method . . . . . . . 134 4.6 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . 136 4.6.1 A problem on an orthogonal grid . . . . . . . . . . . . . . 137 4.6.2 A problem on a non-orthogonal grid . . . . . . . . . . . . 139
contents xi 5 Locally linearized mimetic domain decomposition methods 141 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.2 Spatial semidiscretization: an interpolating MFD method . . . . 143 5.3 Time integration: a locally linearized FSRKm+1 method . . . . . 148 5.3.1 Domain decomposition operator splitting with local linearization . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.3.2 The locally linearized FSRKm+1 method . . . . . . . . . . 151 5.3.3 A first-order unconditionally stable method . . . . . . . . 153 5.3.4 A second-order conditionally stable method . . . . . . . . 155 5.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . 158 5.4.1 A problem on a pseudo-random grid . . . . . . . . . . . . 158 5.4.2 A problem on a smooth grid . . . . . . . . . . . . . . . . 160 5.4.3 A problem on a parametric grid . . . . . . . . . . . . . . . 162 6 Concluding remarks 169 Resumen 175 Bibliography 183
Page 1: TESIS DOCTORAL Mimetic Fractional S
Page 5: A mi breve familia Y a Laura, por t
Page 8 and 9: viii agradecimientos apoyado en mis
Page 13 and 14: 1 Introduction This thesis is devot
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Resumen La presente memoria se enma
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esumen 177 cias finitas centradas e
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184 bibliography U.M. Ascher, S.J.
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186 bibliography Y. Chen, Y. Huang,
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188 bibliography K.J. in ’t Hout
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190 bibliography A. Quarteroni and
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192 bibliography P.N. Vabishchevich
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