Full metadata record
| DC Field | Value | Language |
|---|---|---|
| dc.contributor | Department of Applied Mathematics | en_US |
| dc.contributor.advisor | Lin, Yanping (AMA) | en_US |
| dc.contributor.advisor | Li, Buyang (AMA) | en_US |
| dc.creator | Rao, Qiqi | - |
| dc.identifier.uri | https://theses.lib.polyu.edu.hk/handle/200/14149 | - |
| dc.language | English | en_US |
| dc.publisher | Hong Kong Polytechnic University | en_US |
| dc.rights | All rights reserved | en_US |
| dc.title | Numerical analysis of subdiffusion, Navier-Stokes equations, and fluid-structure interaction | en_US |
| dcterms.abstract | This thesis is devoted to high-order convergent numerical methods for some nonlinear parabolic equations with rough initial data and evolving domains. | en_US |
| dcterms.abstract | Chapter 2 and Chapter 3 are devoted to the analysis of numerical methods for subdiffusion and Navier-Stokes (NS) equations with rough data. In Chapter 2, a new spectral method is constructed for the linear and semilinear subdiffusion equations with possibly discontinuous rough initial data. The new method effectively combines several computational techniques, including the contour integral representation of the solutions, the quadrature approximation of contour integrals, the exponential integrator using the de la Vallée Poussin (VP) means of the source function, and a decomposition of the time interval geometrically refined towards the singularity of the solution and the source function. Rigorous error analysis shows that the proposed method has spectral convergence for the linear and semilinear subdiffusion equations with bounded measurable initial data and possibly singular source functions under the natural regularity of the solutions. | en_US |
| dcterms.abstract | Chapter 3 concerns the numerical solution of the two-dimensional NS equations with nonsmooth initial data in the L² space, which is the critical space for the two-dimensional NS equations to be well-posed. In this case, the solutions of the NS equations exhibit certain singularities at t = 0, e.g., the Hs norm of the solution blows up as t → 0 when s > 0. To date, the best convergence result proved in the literature are first-order accuracy in both time and space for the semi-implicit Euler time-stepping scheme and divergence-free finite elements (even high-order finite elements are used), while numerical results demonstrate that second-order convergence in time and space may be achieved. Therefore, there is still a gap between numerical analysis and numerical computation for the NS equations with L² initial data. The primary challenge to realizing high-order convergence is the insufficient regularity in the solutions due to the rough initial condition and the nonlinearity of the equations. In this work, we propose a fully discrete numerical scheme that utilizes the Taylor-Hood or Stokes-MINI finite element method (FEM) for spatial discretization and an implicit-explicit Runge-Kutta (RK) time-stepping method in conjunction with graded stepsizes. By employing discrete semigroup techniques, sharp regularity estimates, negative norm estimates and the L² projection onto the divergence-free Raviart-Thomas (RT) element space, we prove that the proposed scheme attains second-order convergence in both space and time. Numerical examples are presented to support the theoretical analysis. In particular, the convergence in space is at most second order even higher-order finite elements are used. This shows the sharpness of the convergence order proved in this chapter. | en_US |
| dcterms.abstract | Chapter 4, Chapter 5 and Chapter 6 are devoted to the high-order convergent FEM for parabolic equations in evolving domains. | en_US |
| dcterms.abstract | As a specific type of shape gradient descent algorithm, shape gradient flow is widely used for shape optimization problems constrained by partial differential equations (PDEs). In this approach, the constraint PDEs could be solved by finite element methods on a domain with a solution-driven evolving boundary. Rigorous analysis for the stability and convergence of such finite element approximations is still missing from the literature due to the complex nonlinear dependence of the boundary evolution on the solution. In Chapter 4, rigorous analysis of numerical approximations to the evolution of the boundary in a prototypical shape gradient flow is addressed. First-order convergence in time and k-th order convergence in space for finite elements of degree k ≥ 2 are proved for a linearly semi-implicit evolving finite element algorithm up to a given time. The theoretical analysis is consistent with the numerical experiments, which also illustrate the effectiveness of the proposed method in simulating two- and three-dimensional boundary evolution under shape gradient flow. The extension of the formulation, algorithm and analysis to more general shape density functions and constraint PDEs is also discussed. | en_US |
| dcterms.abstract | In Chapter 5, the numerical solution of the Stokes equations on an evolving domain with a moving boundary is studied based on the arbitrary Lagrangian-Eulerian (ALE) finite element method along the trajectories of the evolving mesh. The error of the semidiscrete ALE method is shown to be O(hr+1) for velocity in L∞(0, T; L²) norm and O(hr) for pressure in L²(0, T; L²) norm by employing the Taylor-Hood finite elements of degree r ≥ 2 , using Nitsche's duality argument adapted to an evolving mesh, by proving that the material derivative and the Stokes-Ritz projection commute up to terms which have optimal-order convergence in the L² norm. Numerical examples are provided to support the theoretical analysis. | en_US |
| dcterms.abstract | In Chapter 6, the convergence of ALE-FEMs for fluid-structure interaction (FSI) problems with a solution-driven moving interface is studied. In addition to the finite element approximation errors, the geometric approximation errors due to the unknown interface motion and mesh evolvement are considered as well within the ALE frame. By adding an initial correction term to the numerical scheme, the optimal convergence rates of fluid velocity, structural displacement and ALE mesh motion in L∞(0, T; H¹) norm, as well as of pressure in L∞(0, T; L²) norm, are proved for a fully discrete, monolithic FEM which tracks the unknown moving interface using the ALE approach. Numerical experiments in both two and three dimensions are presented to support the theoretical error analysis. | en_US |
| dcterms.extent | xii, 187 pages : color illustrations | en_US |
| dcterms.isPartOf | PolyU Electronic Theses | en_US |
| dcterms.issued | 2025 | en_US |
| dcterms.educationalLevel | Ph.D. | en_US |
| dcterms.educationalLevel | All Doctorate | en_US |
| dcterms.accessRights | open access | en_US |
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