Projected Gradient stabilization for unfitted finite element methods with application to tumor growth

Abstract

Motivated by mathematical models for tumor growth, the work conducted in this thesis is focused on stabilization techniques for unfitted finite element methods (FEMs). In such models, appropriate transmission conditions are imposed on the possibly evolving sharp interface between the subdomains occupied by the tumor and surrounding tissue. To avoid frequent remeshing, an unfitted FEM may be employed that uses a fixed background mesh together with an implicit description of the geometry. However, the presence of small cut cells can lead to numerical instabilities, poor conditioning of the system matrix and loss of accuracy caused by small cut cells. To deal with this issue, we introduce a new ghost penalty based on the difference between two consistent discretizations of the Laplacian operator. The proposed projected-gradient stabilization is straightforward to implement and provides an implicit extension of the solution beyond the physical domain. We show that the bilinear form of the stabilization term is symmetric and establish second order convergence in $L^2$ for the solution of the discrete problem. To overcome difficulties associated with numerical integration over sharp embedded interfaces, a diffuse-interface description based on a level set representation is developed. Results of several numerical examples support the theoretical analysis and illustrate the performance of the proposed unfitted FEM. Since the lumped-mass $L^2$ projection that we use for gradient recovery is at most second-order accurate, we introduce nodal averaging as an alternative projection operator for the stabilization term to attain optimal-order accuracy for higher-order polynomial approximations. The stabilization concept is then extended to unfitted FEMs for elliptic interface problems with discontinuous coefficients, for which analogous stability and convergence results are obtained. Moreover, the proposed stabilization is incorporated into an unfitted FEM for a convection-diffusion problem with an embedded interface, and its effectiveness is demonstrated by a numerical example. The unfitted FEM with projected-gradient stabilization is applied to a mathematical model for tumor growth. Using formal asymptotic expansions, a thin-rim limit problem for a tumor growth model is derived. In the thin-rim limit, the pressure satisfies a Poisson equation with a Robin boundary condition in a time-dependent domain whose evolution is governed by a forced mean curvature flow. In the case of stationary, rotationally symmetric solutions, the weak-star convergence of the pressure solution in $L^\infty$ is proven. A generalized thin-rim limit problem is discretized using the proposed stabilized unfitted FEM. The obtained numerical results exhibit good qualitative agreement with results published in the literature and illustrate convergence properties of the proposed method.