We propose a novel unfitted finite element method based on polynomial splines over hierarchical T-meshes (PHT-splines) for elliptic partial differential equations (PDEs) posed on domains whose boundaries are described implicitly by a level-set function. Exploiting the enhanced smoothness of the spline space, we formulate the method by minimizing the $L^2$ residual of the PDE over an extended neighborhood of the physical domain. This formulation allows the method to be implemented entirely on the background mesh using standard element-wise quadrature. It yields a symmetric positive-definite linear system and avoids the geometric construction of curved cut subregions in boundary elements, as well as the associated nontrivial numerical integration. We establish well-posedness of the numerical scheme and derive optimal \emph{a priori} error estimates in the $H^1$ norm. In addition, we employ a residual-based \emph{a posteriori} error indicator to guide local adaptive refinement of the hierarchical T-meshes. Numerical experiments corroborate the theoretical results and demonstrate the accuracy and efficiency of the proposed approach.
In this work, we propose an unfitted finite element scheme to approximate the solution of the heat equation on moving domains. We use the {\phi}-FEM paradigm in which the computational domain is described implicitly by a level-set function {\phi}. This function is incorporated into the variational formulation in order...
Michel Duprez, Johan Marguet, Alexei Lozinski et al.· 0 citations
This paper is the first in a three-part series on the construction of polynomial splines with the highest order of smoothness over hierarchical T-meshes, referred to as $\PHtwoT$-splines. For splines of bi-degree $(d,d)$, we study suitable refinement conditions for the subsequent basis construction, which requires dime...
In this work, we propose and analyze low-order nonconforming face-based finite element methods for elliptic problems on complex hybrid three-dimensional meshes, composed of arbitrary polytopal cells.
The focus is on scalar problems, but the approach can be extended in a tensor manner to design generalizations of the cl...
A. Brunel, J. Latché· ESAIM: Mathematical Modellin...· 0 citations
We develop a localized hybrid finite element--random feature method (FEM--RFM) for elliptic interface problems on unfitted background meshes. The central idea is to use the random feature method (RFM) only in the narrow band formed by interface-cut cells of a uniform Cartesian background mesh,while retaining standard $...
A convergence theory for geometric multigrid with vertex-patch smoothers applied to cut finite element discretizations of the Poisson problem is developed and two-level convergence bounds uniform in the mesh size and the cut geometry, and W-cycle bounds under an additional smallness assumption on the two-level rate are...
Michał Wichrowski· 3 citations
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