arXiv subjects

Raphaël Bulle

Publications and source records attributed to Raphaël Bulle.

5 recordsLinked to original sources

arXiv2026

A multigrid and neural network approach to reduce the computational cost of phi-FEM

In this work, we present a combination of a multigrid approach and the phi-FEM immersed boundary finite element method to reduce its computational cost while preserving its accuracy. To further reduce the numerical cost of the approach, we also propose the combination of the previous technique with some neural network methods. We illustrate the efficiency of these two approaches with numerical test cases in 2D and 3D.

math.NA↗

arXiv2026

A penalized ϕ-FEM scheme for the Poisson Dirichlet problem

In this work, we analyze a penalized variant of the ϕ-FEM scheme for the Poisson equation with Dirichlet boundary conditions. The ϕ-FEM is a recently introduced unfitted finite element method based on a level-set description of the geometry, which avoids the need for boundary-fitted meshes. Unlike the original ϕ-FEM formulation, the method proposed here enforces boundary conditions through a penalization term. This approach has the advantage that the level-set function is required only on the cells adjacent to the boundary in the variational formulation. The scheme is stabilized using a ghost penalty technique. We derive a priori error estimates, showing optimal convergence in the H1 semi-norm and quasi-optimal convergence in the L2 norm under suitable regularity assumptions. Numerical experiments are presented to validate the theoretical results and to compare the proposed method with both the original ϕ-FEM and the standard fitted finite element method.

math.NA↗

arXiv2026

An adaptive multimesh rational approximation scheme for the spectral fractional Laplacian

We propose a novel multimesh rational approximation scheme for the numerical solution of the (homogeneous) Dirichlet problem for the spectral fractional Laplacian. The scheme combines a rational approximation of the function $λ\mapsto λ^{-s}$ with a family of finite element discretizations of parameter-dependent, non-fractional partial differential equations (PDEs). The key idea that underpins the proposed scheme is that each parametric PDE is numerically solved on an individually tailored finite element mesh. This is in contrast to existing single-mesh approaches that employ the same finite element mesh across all parametric PDEs. We develop an a posteriori error estimation strategy for the proposed rational approximation scheme and design an adaptive multimesh refinement algorithm. Numerical experiments demonstrate that our adaptive multimesh approach achieves faster convergence rates than uniform mesh refinement and yields significant reductions in computational costs -- both in terms of the overall number of degrees of freedom and the actual runtime -- when compared to the corresponding adaptive algorithm in a single-mesh setting.

math.NA↗

arXiv2023

An a posteriori error estimator for the spectral fractional power of the Laplacian

We develop a novel a posteriori error estimator for the $L^2$ error committed by the finite element discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes advantage of the semi-discretization scheme using rational approximations which allow to reformulate the fractional problem into a family of non-fractional parametric problems. The estimator involves applying the implicit Bank-Weiser error estimation strategy to each parametric non-fractional problem and reconstructing the fractional error through the same rational approximation used to compute the solution to the original fractional problem. In addition we propose an algorithm to adapt both the finite element mesh and the rational scheme in order to balance the discretization errors. We provide several numerical examples in both two and three-dimensions demonstrating the effectivity of our estimator for varying fractional powers and its ability to drive an adaptive mesh refinement strategy.

math.NA↗

arXiv2022

Hierarchical a posteriori error estimation of Bank-Weiser type in the FEniCS Project

In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp.283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element mesh. Despite the promise of Bank-Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, they have seen little use in practical computational problems. The focus of this contribution is to describe a novel implementation of hierarchical estimators of the Bank-Weiser type in a modern high-level finite element software with automatic code generation capabilities. We show how to use the estimator to drive (goal-oriented) adaptive mesh refinement and to mixed approximations of the nearly-incompressible elasticity problems. We provide comparisons with various other used estimators. An open-source implementation based on the FEniCS Project finite element software is provided as supplementary material.

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