Publicação
Keller-Segel-Flow systems: analytical and numerical study
| Resumo: | In the present work, we study a system of partial differential equations defined by two parabolic equations, that can be seen as natural extensions of the classical Keller–Segel system, an elliptic equation derived from Darcy’s Law and a hyperbolic equation based on stress–strain relations. In this system, the diffusion coefficients and reaction terms of the parabolic equations depend on their respective solutions, while the convection terms are influenced by the solutions of the elliptic, parabolic and hyperbolic equations. This coupled system can be used to mathematically describe the complex interplay of cell migration in response to chemical gradients, interstitial fluid flow and tissue displacement. Specifically, the parabolic equations model the dynamics of cell density and chemical signal, the elliptic equation captures interstitial fluid flow via Darcy’s Law and the hyperbolic equation describes the displacement of the medium resulting from viscoelastic response.Beginning with the simplified case excluding the hyperbolic and elliptic component, we propose a numerical method to solve this system. We then extend this method to incorporate the elliptic component and then we consider the fully coupled problem in a two-dimensional spatial domain. The proposed numerical schemes may be interpreted both as fully discrete piecewise linear finite element methods and as finite difference methods on non-uniform grids. We provide theoretical results supporting the convergence of our numerical schemes. Specifically, we prove second-order convergence in space with respect to a discrete $H^1$-norm and optimal first-order convergence in time with respect to a discrete $L^2$-norm. These results are established under weaker regularity assumptions than those commonly required in the literature when Taylor formula is used in the convergence analysis.Our convergence results can be seen as supraconvergence and superconvergence results. In fact, from a finite difference perspective, our truncation error is of first order in the $\|\cdot\|_\infty$-norm. Simultaneously, in the finite element methods context, where first-order convergence in the $H^1$-norm is standard for piecewise linear elements, our second-order result constitutes a superconvergence result. We also establish stability for the proposed methods. In addition to the theoretical results, we present numerical experiments that illustrate the convergence behavior. The numerical simulation of cell migration in different scenarios is also included in the present work. |
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| Autores principais: | Fernandes, Augusto Manuel de Oliveira |
| Assunto: | métodos numéricos Sistema Keller-Segel, quimiotaxia fluxo de fluído intersticial supra-superconvergência viscoelasticidade Keller Segel systems, chemotaxis interstitial fluid flow numerical methods supra-superconvergence viscoelasticity |
| Ano: | 2025 |
| País: | Portugal |
| Tipo de documento: | tese de doutoramento |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade de Coimbra |
| Idioma: | inglês |
| Origem: | Estudo Geral - Universidade de Coimbra |
| Resumo: | In the present work, we study a system of partial differential equations defined by two parabolic equations, that can be seen as natural extensions of the classical Keller–Segel system, an elliptic equation derived from Darcy’s Law and a hyperbolic equation based on stress–strain relations. In this system, the diffusion coefficients and reaction terms of the parabolic equations depend on their respective solutions, while the convection terms are influenced by the solutions of the elliptic, parabolic and hyperbolic equations. This coupled system can be used to mathematically describe the complex interplay of cell migration in response to chemical gradients, interstitial fluid flow and tissue displacement. Specifically, the parabolic equations model the dynamics of cell density and chemical signal, the elliptic equation captures interstitial fluid flow via Darcy’s Law and the hyperbolic equation describes the displacement of the medium resulting from viscoelastic response.Beginning with the simplified case excluding the hyperbolic and elliptic component, we propose a numerical method to solve this system. We then extend this method to incorporate the elliptic component and then we consider the fully coupled problem in a two-dimensional spatial domain. The proposed numerical schemes may be interpreted both as fully discrete piecewise linear finite element methods and as finite difference methods on non-uniform grids. We provide theoretical results supporting the convergence of our numerical schemes. Specifically, we prove second-order convergence in space with respect to a discrete $H^1$-norm and optimal first-order convergence in time with respect to a discrete $L^2$-norm. These results are established under weaker regularity assumptions than those commonly required in the literature when Taylor formula is used in the convergence analysis.Our convergence results can be seen as supraconvergence and superconvergence results. In fact, from a finite difference perspective, our truncation error is of first order in the $\|\cdot\|_\infty$-norm. Simultaneously, in the finite element methods context, where first-order convergence in the $H^1$-norm is standard for piecewise linear elements, our second-order result constitutes a superconvergence result. We also establish stability for the proposed methods. In addition to the theoretical results, we present numerical experiments that illustrate the convergence behavior. The numerical simulation of cell migration in different scenarios is also included in the present work. |
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