CONTENTS -- Preface -- Generalized Riemann Problems: From the Scalar Equation to Multidimensional Fluid Dynamics M. Ben-Arlizi and J. Falcovitz -- 1. Introduction -- 2. GRP Scheme - Basic Properties -- 3. Self-Similar Solutions in 2-D -- 4. The Analysis of 2-D Riemann-Type Solutions -- 5. The 2-D Guckenheimer Equation -- 6. Euler Equations of Quasi I-D Compressible Inviscid Flow -- 7. The GRP for Quasi l-D Compressible Inviscid Flow -- 8. The GRP Numerical Method for Quasi I-D Compressible Inviscid Flow -- 9. Fluid Dynamical Examples -- 9.1. Wave Dynamics in a Duct with a Converging Segment -- 9.2. Shock diffraction by a square cavity -- 9.3. Second reflection of a shock wave by a double wedge -- 9.4. Lagrange-Euler Scheme for a Gas-Grains Mixture -- References -- Some Observations on a Multiscale Finite Element Method for Problems with Variables Separated Coefficients S. Jiang and Y-Q. Huang -- 1. Introduction -- 2. Formulations -- 2.1. Model problem -- 2.2. Remarks -- 3. Base functions under variables separated condition -- 4. Numerical experiment -- 4.1. Implementation and explanation -- 4.2. Experiment results -- References -- A Finite Element Method for the Simulation of a Liquid Droplet Impinging on a Solid Surface S. Ganesan and L. Tobiska -- 1. Introduction -- 2. Mathematical model -- 2.1. Governing equations -- 2.2. Initial and boundary conditions -- 2.3. Dimensionless form -- 3. Numerical Scheme -- 3.1. Arbitrary Lagrangian Eulerian (ALE) Approach -- 3.2. Variational form -- 3.3. Discretisation in time and linearization -- 3.4. Discretisation in space -- 3.5. Solving the discrete problems -- 3.6. Free surface update -- 3.7. Automatic mesh moving techniques -- 4. Numerical results -- Acknowledgement -- References.

Applications of Moving Mesh Methods to the Fourier Spectral Approximations of Phase-Field Equations P. Yu, L. Q. Chen and Q. Du -- 1. Introduction -- 2. Moving mesh methods -- 3. Variational Moving Mesh PDEs and their Fourier Spectral Implementation -- 3.1. A computational domain representation -- 3.2. A physical domain representation -- 3.3. Fourier-Spectral Implementation of MMPDEs -- 3.4. Comparisons of two representations -- 4. Adaptive Spectral Methods for Phase-Field Equations -- 4.1. Allen-Cahn equations -- 4.2. Numerical examples for Allen- Cahn equations -- 5. Extension and Further Discussion -- 5.1. Extension to Cahn-Hilliard equation -- 5.2. Extension to Phase Field Inhomogeneous Elasticity Equations -- 5.3. Switching on the Mesh Adaptation -- 5.4. Further discussion -- Acknowledgments -- References -- On the Comparison of Evolution Galerkin and Discontinuous Galerkin Schemes K. Baumbach and M. Lukacova-Medvidova -- 1. Introduction -- 2. Finite volume evolution Galerkin methods -- 3. Discontinuous Galerkin methods -- 4. Numerical experiments -- 4.1. Shallow Water Equations -- 4.2. Dam break problem -- 4.3. Accuracy and efficiency tests -- 5. Conclusion -- Acknowledgement -- References -- Time-Domain Finite Element Methods for Maxwell's Equations in Dispersive Media: A Review J. Li and Y Chen -- 1. Introduction -- 2. Maxwell's equations and dispersive media -- 3. Finite element methods for dispersive media -- 3.1. Standard error analysis -- 3.2. Superconvergence analysis -- 4. Conclusion and further issues -- 4.1. Regularity -- 4.2. Hp-adaptive methods -- 4.3. A posteriori error estimation -- 4.4. Solvers -- 4.5. Perfectly matched layer (PML) -- References -- Superconvergence and A Posteriori Error Estimates of Nonconforming FEM for Boundary Control Governed by Stokes Equations H. Liu and N. Yan -- 1. Introduction.

2. Finite element approximation of boundary control problems -- 3. Superconvergence analysis and recovery for the control u -- 4. Superconvergence analysis for state and co-state and recovery type a posteriori error estimates -- 5. Numerical examples -- 6. Discussions -- Acknowledgments -- References -- Optimal Superconvergent One Step Quadratic Spline Collocation Methods for Helmholtz Problems B. Bialecki, G. Fairweather, A. K arageorghis and Q. N. Nguyen -- 1. Introduction -- 2. Optimal Quadratic Spline Collocation Methods for a Two-Point Boundary Value Problem -- 2.1. Preliminaries -- 2.2. Optimal Quadratic Spline Collocation Methods -- 2.3. Numerical Results -- 3. Helmholtz Problem -- 3.1. New Optimal Quadratic Spline Collocation Method -- 3.2. Numerical Results -- 4. Extensions and Generalizations -- Acknowledgment -- References -- Some Progress on Superconvergence for Mixed Finite Element Methods S. Jia, H. Xie and X Yin -- 1. Introdution -- 2. Operator Expressions -- 2.1. The Eigenvalue Error Expansion for The First Type of Eigenvalue Problems -- 2.2. The Eigenvalue Error Expansion for The Second Type of Eigenvalue Problems -- 3. Mixed FEMs for Second Order Elliptic Eigenvalue Problems -- 3.1. Two Dimensional Rectangular RTo Element -- 3.2. Two Dimensional Rectangular RTk Element -- 3.3. Two Dimensional Rectangular BDF Mk Element -- 3.4. Three Dimensional RTo Element -- 4. Mixed FEMs for Biharmonic Eigenvalue Problems -- 4.1. Q1 Element -- 4.2. Qk Element -- 5. Mixed FEMs for Stokes Eigenvalue Problems -- 5.1. B - R Rectangular Element -- 5.2. Q2 - P 1 Rectangular Element -- 5.3. Hood-Taylor Rectangular Element -- 5.4. Q1 - Qo Rectangular Element -- 5.5. Hood-Taylor Triangular Element -- 5.6. PI - PI Triangular Element -- 5.7. Another Form for Stokes Problems -- 6. Concluding Remarks -- Acknowledgments -- References.

Convergence of Green Iterations for Schrodinger Equations M. J. Mohlenkamp and T. Young -- 1. Green Function Iterations -- 2. Background Material -- 3. Proof of Main Result -- 4. Concluding Remarks -- References -- Convolution and Wiener Amalgam Spaces on the Affine Group C. Heil and G. Kutyniok -- 1. Introduction -- 2. Notation and Preliminary Results -- 2.1. General Notation -- 2.2. Preliminary Lemmas -- 2.3. Lemmas on Convolution -- 2.4. Amalgam Spaces on the Affine Group -- 3. Amalgam Spaces and Convolution on the Affine Group -- Acknowledgments -- References -- An Approximation Formula in Hilbert Space Z. Zhang and N. Saito -- 1. Introduction -- 2. A new frame approximation operator (IT -- { ('))1n -- 3. Approximation by (0": -- [ ('»rrt -- References -- The Gibbs Phenomenon: A Digression in Wavelet Hybrid Sampling Series X A. Shen -- 1. Introduction -- 2. Sampling theory based on wavelets -- 2.1. Sampling functions associated with scaling functions -- 2.2. Sampling and hybrid sampling series in scaling spaces -- 3. The Gibbs phenomenon in hybrid sampling series -- 3.1. Abel mean for wavelet with compact support -- 3.2. The positive hybrid sampling series -- 4. Numerical examples -- 5. Summary -- Acknowledgment -- References -- A Unified Framework for Segment ation-Assisted Image Registration J. Liu and W. Yang -- 1. Introduction and Related Work -- 2. Segmentation Guided Registration Model -- 2.1. Frameworks based on intensity homegeneity -- 2.2. Level Set Formulation of the LC-based model -- 2.3. A SSD-based Segmentation + Registration Model -- 3. Experimental Results -- 3.1. Registration based on the Segmentation + Registration Component -- 3.2. Registrations of our Segmentation + Registration LC model -- 4. Conclusions -- References -- Geometric Hermite Interpolation with Pythagorean Hodograph Cubics X-A. Han, Y-C. Ma and H.- Y Sun.

1. Introduction -- 2. Geometric Hermite interpolation with a PH cubic -- 3. Examples -- References -- Shape Gradient for the Stokes Flow by Dierentiability of a Saddle Point Z. Gao and Y Ma -- 1. Introduction -- 2. Formulation of the problem -- 3. The Velocity Method -- 4. A Saddle Point Formulation -- 5. Function Space Parametrization -- 6. Function Space Embedding -- Acknowledgments -- References -- Numerical Simulation of Flow Over Two Rotating Self-Moving Circular Cylinders S. Sungnul and N. P. M oshkin -- 1. Introduction -- 2. Mathematical Formulation -- 3. Numerical Algorithm -- 4. Validation of Numerical Algorithm -- 5. Results and Discussion -- 5.1. Streamlines patterns depending on the Reynolds number and rate of rotation -- 5.2. Streamline patterns depending on the gap spacing and the rate of rotation -- 6. Conclusion -- References -- A Note on PPT Forms and Separability of Multipartite States X.-H. Wang, S.-M. Fei and K. Wu -- 1. Introduction -- 2. Main results -- 3. Examples -- 4. Discussions -- References -- A Structured Data Least Squares Algorithm and Its Application in Digital Filtering H. Guo and R. Renaut -- 1. Introduction -- 2. An Algorithm for Structured Data Least Squares -- 2.1. Properties of the SDLS problem -- 2.2. Algorithm Description -- 2.3. Numerical approaches for solving the iteration equation -- 2.3.1. solution of B 18z = f - B! 8),: -- 2.3.2. Solution of B2Bli B! 8), = B2Bl i f - r -- 3. Experiments -- 4. Conclusion -- Appendix A. Algorithm for Data Least Squares -- References -- Short Implementation of Bisection in Matlab L. Chen -- 1. Introduction -- 2. Bisections in adaptive finite element methods -- 2.1. Newest vertex bisection -- 2.2. Longest edge bisection -- 3. Implementation -- 3.1. Data structure -- 3.2. Initial labeling -- 3.3. Marking strategy -- 3.4. Refinement -- 4. Numerical example.

Appendix A: Matlab code for bisection.