arXiv subjects

Hridya Dilip

Publications and source records attributed to Hridya Dilip.

3 recordsLinked to original sources

arXiv2026

High order ghost-FEM for incompressible Navier-Stokes equations on moving domains

We develop a new numerical technique for approximating solutions of the Navier-Stokes equations on moving domains. The method aims at simulating an incompressible fluid past an object whose motion is assigned a priori using a level-set function. The proposed approach relies on a space discretization based on the ghost finite element method (ghost-FEM), which allows computations on unfitted meshes and avoids costly remeshing as the domain evolves in time. Time integration is performed using an IMplicit-EXplicit (IMEX) scheme to address the nonlinearity of the convective term, ensuring high-order accuracy for incompressible flows. The error introduced by the geometrical approximation is handled using the Shifted Boundary Method, which allows higher order approximations of boundary conditions on unfitted meshes. Dirichlet boundary conditions are imposed weakly by means of Nitsche's method. The associated stabilization parameter is chosen by solving a generalized eigenvalue problem, ensuring stability and accuracy of the numerical scheme. We present a series of numerical experiments designed to validate the accuracy of the proposed method, as well as comparisons with established benchmark problems involving moving boundaries.

math.NA↗

arXiv2025

Multigrid methods for the ghost finite element approximation of elliptic problems

We present multigrid methods for solving elliptic partial differential equations on arbitrary domains using the nodal ghost finite element method, an unfitted boundary approach where the domain is implicitly defined by a level-set function. This method achieves second-order accuracy and offers substantial computational advantages over both direct solvers and finite-difference-based multigrid methods. A key strength of the ghost finite element framework is its variational formulation, which naturally enables consistent transfer operators and avoids residual splitting across grid levels. We provide a detailed construction of the multigrid components in both one and two spatial dimensions, including smoothers, transfer operators, and coarse grid operators. The choice of the stabilization parameter plays a crucial role in ensuring well-posedness and optimal convergence of the multigrid method. We derive explicit algebraic expressions for this parameter based on the geometry of cut cells. In the two-dimensional setting, we further improve efficiency by performing additional smoothing exclusively on cut cells, reducing computational cost without compromising convergence. Numerical results validate the proposed method across a range of geometries and confirm its robustness and scalability.

math.NA↗

arXiv2023

Space-time unfitted finite element methods for time-dependent problems on moving domains

We propose a space-time scheme that combines an unfitted finite element method in space with a discontinuous Galerkin time discretisation for the accurate numerical approximation of parabolic problems with moving domains or interfaces. We make use of an aggregated finite element space to attain robustness with respect to the cut locations. The aggregation is performed slab-wise to have a tensor product structure of the space-time discrete space, which is required in the numerical analysis. We analyse the proposed algorithm, providing stability, condition number bounds and anisotropic \emph{a priori} error estimates. A set of numerical experiments confirm the theoretical results for a parabolic problem on a moving domain. The method is applied for a mass transfer problem with changing topology.

math.NA↗