No Arabic abstract
This paper proposes an interface/boundary-unfitted eXtended hybridizable discontinuous Galerkin (X-HDG) method for Darcy-Stokes-Brinkman interface problems in two and three dimensions. The method uses piecewise linear polynomials for the velocity approximation and piecewise constants for both the velocity gradient and pressure approximations in the interior of elements inside the subdomains separated by the interface, uses piecewise constants for the numerical traces of velocity on the inter-element boundaries inside the subdomains, and uses piecewise constants or linear polynomials for the numerical traces of velocity on the interface. Optimal error estimates are derived for the interface-unfitted X-HDG scheme. Numerical experiments are provided to verify the theoretical results and the robustness of the proposed method.
An interface/boundary-unfitted eXtended hybridizable discontinuous Galerkin (X-HDG) method of arbitrary order is proposed for linear elasticity interface problems on unfitted meshes with respect to the interface and domain boundary. The method uses piecewise polynomials of degrees $k (>= 1)$ and $k-1$ respectively for the displacement and stress approximations in the interior of elements inside the subdomains separated by the interface, and piecewise polynomials of degree $k$ for the numerical traces of the displacement on the inter-element boundaries inside the subdomains and on the interface/boundary of the domain. Optimal error estimates in $L^2$-norm for the stress and displacement are derived. Finally, numerical experiments confirm the theoretical results and show that the method also applies to the case of crack-tip domain.
In this paper, a stabilized extended finite element method is proposed for Stokes interface problems on unfitted triangulation elements which do not require the interface align with the triangulation. The velocity solution and pressure solution on each side of the interface are separately expanded in the standard nonconforming piecewise linear polynomials and the piecewise constant polynomials, respectively. Harmonic weighted fluxes and arithmetic fluxes are used across the interface and cut edges (segment of the edges cut by the interface), respectively. Extra stabilization terms involving velocity and pressure are added to ensure the stable inf-sup condition. We show a priori error estimates under additional regularity hypothesis. Moreover, the errors {in energy and $L^2$ norms for velocity and the error in $L^2$ norm for pressure} are robust with respect to the viscosity {and independent of the location of the interface}. Results of numerical experiments are presented to {support} the theoretical analysis.
This paper presents and analyzes an immersed finite element (IFE) method for solving Stokes interface problems with a piecewise constant viscosity coefficient that has a jump across the interface. In the method, the triangulation does not need to fit the interface and the IFE spaces are constructed from the traditional $CR$-$P_0$ element with modifications near the interface according to the interface jump conditions. We prove that the IFE basis functions are unisolvent on arbitrary interface elements and the IFE spaces have the optimal approximation capabilities, although the proof is challenging due to the coupling of the velocity and the pressure. The stability and the optimal error estimates of the proposed IFE method are also derived rigorously. The constants in the error estimates are shown to be independent of the interface location relative to the triangulation. Numerical examples are provided to verify the theoretical results.
We design an adaptive unfitted finite element method on the Cartesian mesh with hanging nodes. We derive an hp-reliable and efficient residual type a posteriori error estimate on K-meshes. A key ingredient is a novel hp-domain inverse estimate which allows us to prove the stability of the finite element method under practical interface resolving mesh conditions and also prove the lower bound of the hp a posteriori error estimate. Numerical examples are included.
In this paper, we propose a deep unfitted Nitsche method for computing elliptic interface problems with high contrasts in high dimensions. To capture discontinuities of the solution caused by interfaces, we reformulate the problem as an energy minimization involving two weakly coupled components. This enables us to train two deep neural networks to represent two components of the solution in high-dimensional. The curse of dimensionality is alleviated by using the Monte-Carlo method to discretize the unfitted Nitsche energy function. We present several numerical examples to show the efficiency and accuracy of the proposed method.