A Fixed Mesh Method With Immersed Finite Elements for Solving Interface Inverse Problems

We present a new fixed mesh algorithm for solving a class of interface inverse problems for the typical elliptic interface problems. These interface inverse problems are formulated as shape optimization prob- lems whose objective functionals depend on the shape of the interface. Regardless of the location of the interface, both the governing partial differential equations and the objective functional are discretized optimally, with respect to the involved polynomial space, by an immersed finite element (IFE) method on a fixed mesh. Furthermore, the formula for the gradient of the descritized objective function is de- rived within the IFE framework that can be computed accurately and efficiently through the discretized adjoint procedure. Features of this proposed IFE method based on a fixed mesh are demonstrated by its applications to three representative interface inverse problems: the interface inverse problem with an internal measurement on a sub-domain, a Dirichlet-Neumann type inverse problem whose data is given on the boundary, and a heat dissipation design problem

An inverse problem formulation of the immersed boundary method

We formulate the immersed-boundary method (IBM) as an inverse problem. A control variable is introduced on the boundary of a larger domain that encompasses the target domain. The optimal control is the one that minimizes the mismatch between the state and the desired boundary value along the immersed target-domain boundary. We begin by investigating a na\"ive problem formulation that we show is ill-posed: in the case of the Laplace equation, we prove that the solution is unique but it fails to depend continuously on the data; for the linear advection equation, even solution uniqueness fails to hold. These issues are addressed by two complimentary strategies. The first strategy is to ensure that the enclosing domain tends to the true domain as the mesh is refined. The second strategy is to include a specialized parameter-free regularization that is based on penalizing the difference between the control and the state on the boundary. The proposed inverse IBM is applied to the diffusion, advection, and advection-diffusion equations using a high-order discontinuous Galerkin discretization. The numerical experiments demonstrate that the regularized scheme achieves optimal rates of convergence and that the reduced Hessian of the optimization problem has a bounded condition number as the mesh is refined

Solving Parabolic Moving Interface Problems with Dynamical Immersed Spaces on Unfitted Meshes: Fully Discrete Analysis

Immersed finite element (IFE) methods are a group of long-existing numerical methods for solving interface problems on unfitted meshes. A core argument of the methods is to avoid mesh regeneration procedure when solving moving interface problems. Despite the various applications in moving interface problems, a complete theoretical study on the convergence behavior is still missing. This research is devoted to close the gap between numerical experiments and theory. We present the first fully discrete analysis including the stability and optimal error estimates for a backward Euler IFE method for solving parabolic moving interface problems. Numerical results are also presented to validate the analysis

High-Order Extended Finite Element Methods for Solving Interface Problems

In this paper, we study arbitrary order extended finite element (XFE) methods based on two discontinuous Galerkin (DG) schemes in order to solve elliptic interface problems in two and three dimensions. Optimal error estimates in the piecewise $H^1$-norm and in the $L^2$-norm are rigorously proved for both schemes. In particular, we have devised a new parameter-friendly DG-XFEM method, which means that no "sufficiently large" parameters are needed to ensure the optimal convergence of the scheme. To prove the stability of bilinear forms, we derive non-standard trace and inverse inequalities for high-order polynomials on curved sub-elements divided by the interface. All the estimates are independent of the location of the interface relative to the meshes. Numerical examples are given to support the theoretical results

Error analysis of a diffuse interface method for elliptic problems with Dirichlet boundary conditions

We use a diffuse interface method for solving Poisson's equation with a Dirichlet condition on an embedded curved interface. The resulting diffuse interface problem is identified as a standard Dirichlet problem on approximating regular domains. We estimate the errors introduced by these domain perturbations, and prove convergence and convergence rates in the $H^1$-norm, the $L^2$-norm and the $L^\infty$-norm in terms of the width of the diffuse layer. For an efficient numerical solution we consider the finite element method for which another domain perturbation is introduced. These perturbed domains are polygonal and non-convex in general. We prove convergence and convergences rates in the $H^1$-norm and the $L^2$-norm in terms of the layer width and the mesh size. In particular, for the $L^2$-norm estimates we present a problem adapted duality technique, which crucially makes use of the error estimates derived for the regularly perturbed domains. Our results are illustrated by numerical experiments, which also show that the derived estimates are sharp.Comment: Revised version. In particular, the L2-error analysis for the finite element method has been extende

Robust and parallel scalable iterative solutions for large-scale finite cell analyses

The finite cell method is a highly flexible discretization technique for numerical analysis on domains with complex geometries. By using a non-boundary conforming computational domain that can be easily meshed, automatized computations on a wide range of geometrical models can be performed. Application of the finite cell method, and other immersed methods, to large real-life and industrial problems is often limited due to the conditioning problems associated with these methods. These conditioning problems have caused researchers to resort to direct solution methods, which signifi- cantly limit the maximum size of solvable systems. Iterative solvers are better suited for large-scale computations than their direct counterparts due to their lower memory requirements and suitability for parallel computing. These benefits can, however, only be exploited when systems are properly conditioned. In this contribution we present an Additive-Schwarz type preconditioner that enables efficient and parallel scalable iterative solutions of large-scale multi-level hp-refined finite cell analyses.Comment: 32 pages, 17 figure

Approximation Capabilities of Immersed Finite Element Spaces for Elasticity Interface Problems

We construct and analyze a group of immersed finite element (IFE) spaces formed by linear, bilinear and rotated Q1 polynomials for solving planar elasticity equation involving interface. The shape functions in these IFE spaces are constructed through a group of approximate jump conditions such that the unisolvence of the bilinear and rotated Q1 IFE shape functions are always guaranteed regardless of the Lam\`e parameters and the interface location. The boundedness property and a group of identities of the proposed IFE shape functions are established. A multi-point Taylor expansion is utilized to show the optimal approximation capabilities for the proposed IFE spaces through the Lagrange type interpolation operators

A matched alternating direction implicit (ADI) method for solving the heat equation with interfaces

A novel Douglas alternating direction implicit (ADI) method is proposed in this work to solve a two-dimensional (2D) heat equation with interfaces. The ADI scheme is a powerful finite difference method for solving parabolic equations, due to its unconditional stability and high efficiency. However, it suffers from a serious accuracy reduction in space for interface problems with different materials and nonsmooth solutions. If the jumps in a function and its derivatives are known across the interface, rigorous ADI schemes have been successfully constructed in the literature based on the immersed interface method (IIM) so that the spatial accuracy can be restored. Nevertheless, the development of accurate and stable ADI methods for general parabolic interface problems with physical interface conditions that describe jumps of a function and its flux, remains unsolved. To overcome this difficulty, a novel tensor product decomposition is proposed in this paper to decouple 2D jump conditions into essentially one-dimensional (1D) ones. These 1D conditions can then be incorporated into the ADI central difference discretization, using the matched interface and boundary (MIB) technique. Fast algebraic solvers for perturbed tridiagonal systems are developed to maintain the computational efficiency. Stability analysis is conducted through eigenvalue spectrum analysis, which numerically demonstrates the unconditional stability of the proposed ADI method. The matched ADI scheme achieves the first order of accuracy in time and second order of accuracy in space in all tested parabolic interface problems with complex geometries and spatial-temporal dependent jump conditions

An unfitted hphp-interface penalty finite element method for elliptic interface problems

An $hp$ version of interface penalty finite element method ($hp$-IPFEM) is proposed for elliptic interface problems in two and three dimensions on unfitted meshes. Error estimates in broken $H^1$ norm, which are optimal with respect to $h$ and suboptimal with respect to $p$ by half an order of $p$, are derived. Both symmetric and non-symmetric IPFEM are considered. Error estimates in $L^2$ norm are proved by the duality argument

Inf-sup stability of geometrically unfitted Stokes finite elements

The paper shows an inf-sup stability property for several well-known 2D and 3D Stokes elements on triangulations which are not fitted to a given smooth or polygonal domain. The property implies stability and optimal error estimates for a class of unfitted finite element methods for the Stokes and Stokes interface problems, such as Nitsche-XFEM or cutFEM. The error analysis is presented for the Stokes problem. All assumptions made in the paper are satisfied once the background mesh is shape-regular and fine enough