Immersed Interface Finite Element Methods for Elasticity Interface Problems with Non-Homogeneous Jump Conditions

In this paper,a class of new immersed interface finite element methods (IIFEM) is developed to solve elasticity interface problems with homogeneous and non-homogeneous jump conditions in two dimensions.Simple non-body-fitted meshes are used.For homogeneous jump conditions,both non-conforming and conforming basis functions are constructed in such a way that they satisfy the natural jump conditions. For non-homogeneous jump conditions,a pair of functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface.With such a pair of functions,the discontinuities across the interface in the solution and flux are removed;and an equivalent elasticity interface problem with homogeneous jump conditions is formulated.Numerical examples are presented to demonstrate that such methods have second order convergence.

P_1-nonconforming triangular finite element method for elliptic and parabolic interface problems

The lowest order Pl-nonconforming triangular finite element method （FEM） for elliptic and parabolic interface problems is investigated. Under some reasonable regularity assumptions on the exact solutions, the optimal order error estimates are obtained in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.

A second order isoparametric finite element method for elliptic interface problems

A second order isoparametric finite element method （IPFEM） is proposed for elliptic interface problems. It yields better accuracy than some existing second-order methods, when the coefficients or the flux across the immersed curved interface is discontinuous. Based on an initial Cartesian mesh, a mesh optimization strategy is presented by employing curved boundary elements at the interface, and an incomplete quadratic finite element space is constructed on the optimized mesh. It turns out that the number of curved boundary elements is far less than that of the straight one, and the total degree of freedom is almost the same as the uniform Cartesian mesh. Numerical examples with simple and complicated geometrical interfaces demonstrate the efficiency of the proposed method.

LOCAL DISCONTINUOUS GALERKIN METHOD FOR ELLIPTIC INTERFACE PROBLEMS

In this paper,the minimal dissipation local discontinuous Galerkin method is studied to solve the elliptic interface problems in two-dimensional domains.The interface may be arbitrary smooth curves.It is shown that the error estimates in L;-norm for the solution and the flux are O（h;|log h|）and O(h|log h|;),respectively.In numerical experiments,the successive substitution iterative methods are used to solve the LDG schemes.Numerical results verify the efficiency and accuracy of the method.

A FINITE DIFFERENCE METHOD FOR TWO DIMENSIONAL ELLIPTIC INTERFACE PROBLEMS WITH IMPERFECT CONTACT

In this paper two dimensional elliptic interface problem with imperfect contact is considered,which is featured by the implicit jump condition imposed on the imperfect contact interface,and the jumping quantity of the unknown is related to the flux across the interface.A finite difference method is constructed for the 2D elliptic interface problems with straight and curve interface shapes.Then,the stability and convergence analysis are given for the constructed scheme.Further,in particular case,it is proved to be monotone.Numerical examples for elliptic interface problems with straight and curve interface shapes are tested to verify the performance of the scheme.The numerical results demonstrate that it obtains approximately second-order accuracy for elliptic interface equations with implicit jump condition.

Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems

With the remarkable empirical success of neural networks across diverse scientific disciplines,rigorous error and convergence analysis are also being developed and enriched.However,there has been little theoretical work focusing on neural networks in solving interface problems.In this paper,we perform a convergence analysis of physics-informed neural networks(PINNs)for solving second-order elliptic interface problems.Specifically,we consider PINNs with domain decomposition technologies and introduce gradient-enhanced strategies on the interfaces to deal with boundary and interface jump conditions.It is shown that the neural network sequence obtained by minimizing a Lipschitz regularized loss function converges to the unique solution to the interface problem in H2 as the number of samples increases.Numerical experiments are provided to demonstrate our theoretical analysis.

A Well-Conditioned, Nonconforming Nitsche’s Extended Finite Element Method for Elliptic Interface Problems

In this paper,we introduce a nonconforming Nitsche’s extended finite element method(NXFEM)for elliptic interface problems on unfitted triangulation elements.The solution on each side of the interface is separately expanded in the standard nonconforming piecewise linear polynomials with the edge averages as degrees of freedom.The jump conditions on the interface and the discontinuities on the cut edges(the segment of edges cut by the interface)are weakly enforced by the Nitsche’s approach.In the method,the harmonic weighted fluxes are used and the extra stabilization terms on the interface edges and cut edges are added to guarantee the stability and the well conditioning.We prove that the convergence order of the errors in energy and L 2 norms are optimal.Moreover,the errors are independent of the position of the interface relative to the mesh and the ratio of the discontinuous coefficients.Furthermore,we prove that the condition number of the system matrix is independent of the interface position.Numerical examples are given to confirm the theoretical results.

THE ADAPTIVE IMMERSED INTERFACE FINITE ELEMENT METHOD FOR ELASTICITY INTERFACE PROBLEMS

In this paper, we propose adaptive finite element methods with error control for solving elasticity problems with discontinuous coefficients. The meshes in the methods do not need to fit the interfaces. We establish a residual-based a posteriori error estimate which is λ- independent multiplicative constants; the Lame constant λ steers the incompressibility. The error estimators are then implemented and tested with promising numerical results which will show the competitive behavior of the adaptive algorithm.

High Order Deep Domain Decomposition Method for Solving High Frequency Interface Problems

This paper proposes a high order deep domain decomposition method(HOrderDeepDDM)for solving high-frequency interface problems,which combines high order deep neural network(HOrderDNN)with domain decomposition method(DDM).The main idea of HOrderDeepDDM is to divide the computational domain into some sub-domains by DDM,and apply HOrderDNNs to solve the high-frequency problem on each sub-domain.Besides,we consider an adaptive learning rate annealing method to balance the errors inside the sub-domains,on the interface and the boundary during the optimization process.The performance of HOrderDeepDDM is evaluated on high-frequency elliptic and Helmholtz interface problems.The results indicate that:HOrderDeepDDM inherits the ability of DeepDDM to handle discontinuous interface problems and the power of HOrderDNN to approximate high-frequency problems.In detail,HOrderDeepDDMs(p>1)could capture the high-frequency information very well.When compared to the deep domain decomposition method(DeepDDM),HOrderDeepDDMs(p>1)converge faster and achieve much smaller relative errors with the same number of trainable parameters.For example,when solving the high-frequency interface elliptic problems in Section 3.3.1,the minimum relative errors obtained by HOrderDeepDDMs(p=9)are one order of magnitude smaller than that obtained by DeepDDMs when the number of the parameters keeps the same,as shown in Fig.4.

AN ITERATIVE HYBRIDIZED MIXED FINITE ELEMENT METHOD FOR ELLIPTIC INTERFACE PROBLEMS WITH STRONGLY DISCONTINUOUS COEFFICIENTS

An iterative algorithm is proposed and analyzed based on a hybridized mixed finite element method for numerically solving two-phase generalized Stefan interface problems with strongly discontinuous solutions, conormal derivatives, and coefficients. This algorithm iteratively solves small problems for each single phase with good accuracy and exchange information at the interface to advance the iteration until convergence, following the idea of Schwarz Alternating Methods. Error estimates are derived to show that this algorithm always converges provided that relaxation parameters are suitably chosen. Numeric experiments with matching and non-matching grids at the interface from different phases are performed to show the accuracy of the method for capturing discontinuities in the solutions and coefficients. In contrast to standard numerical methods, the accuracy of our method does not seem to deteriorate as the coefficient discontinuity increases.

COMPUTATIONAL ISSUES IN SENSITIVITY ANALYSIS FOR 1-D INTERFACE PROBLEMS

This paper is concerned with the construction of accurate and efficient computational algorithms for the numerical approximation of sensitivities with respect to a parameter dependent interface location. Motivated by sensitivity analysis with respect to piezoelectric actuator placement on an Euler-Bernonlli beam, this work illustrates the key concepts related to sensitivity equation formulation for interface problems where the parameter of interest determines the location of the interface. A fourth order model problem is considered, and a homogenization procedure for sensitivity computation is constructed using standard finite clement methods. Numerical results show that proper formulation and approximation of the sensitivity interface conditions is critical to obtaining convergent numerical sensitivity approximations. A second order elliptic interface model problem is also mentioned, and the homogenization procedure is outlined briefly for this model.

Immersed Finite Element Method for Interface Problems with Algebraic Multigrid Solver

This article is to discuss the bilinear and linear immersed finite element(IFE)solutions generated from the algebraic multigrid solver for both stationary and moving interface problems.For the numerical methods based on finite difference formulation and a structured mesh independent of the interface,the stiffness matrix of the linear system is usually not symmetric positive-definite,which demands extra efforts to design efficient multigrid methods.On the other hand,the stiffness matrix arising from the IFE methods are naturally symmetric positive-definite.Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and linear IFE methods for both stationary and moving interface problems.The numerical examples demonstrate the features of the proposed algorithms,including the optimal convergence in both L 2 and semi-H1 norms of the IFE-AMG solutions,the high efficiency with proper choice of the components and parameters of AMG,the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems,and the relationship between the cost and the moving interface location.

Deep Unfitted Nitsche Method for Elliptic Interface Problems

This paper proposes a deep unfitted Nitsche method for solving 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 problem involving two weakly coupled components.This enables us to train two deep neural networks to represent two components of the solution in highdimensional space.The curse of dimensionality is alleviated by using theMonte-Carlo method to discretize the unfittedNitsche energy functional.We present several numerical examples to show the performance of the proposed method.

INTERFACE PROBLEMS FOR ELLIPTIC DIFFERENTIAL EQUATIONS

A new approach is given to analyse the regularity of solutions near singular points for the interface problems of second order elliptic partial differential equations. For general equations with nonsymmetric dominant terms and discontinuous piecewise smooth coefficients, it is proved that solutions in H 1 can be docomposed into two parts, one of which is a finite sum of particular solutions to the corresponding homogeneous equations with piecewise constant coefficients, and the other one of which is the regular part. Moreover a priori estimations are proven.

An Augmented Lagrangian Uzawa IterativeMethod for Solving Double Saddle-Point Systems with Semidefinite(2,2)Block and its Application to DLM/FDMethod for Elliptic Interface Problems

.In this paper,an augmented Lagrangian Uzawa iterative method is developed and analyzed for solving a class of double saddle-point systems with semidefinite(2,2)block.Convergence of the iterativemethod is proved under the assumption that the double saddle-point problem exists a unique solution.An application of the iterative method to the double saddle-point systems arising from the distributed Lagrange multiplier/fictitious domain(DLM/FD)finite element method for solving elliptic interface problems is also presented,in which the existence and uniqueness of the double saddle-point system is guaranteed by the analysis of the DLM/FD finite element method.Numerical experiments are conducted to validate the theoretical results and to study the performance of the proposed iterative method.

AN EMBEDDED BOUNDARY METHOD FOR ELLIPTIC AND PARABOLIC PROBLEMS WITH INTERFACES AND APPLICATION TO MULTI-MATERIAL SYSTEMS WITH PHASE TRANSITIONS

The embedded boundary method for solving elliptic and parabolic problems in geometrically complex domains using Cartesian meshes by Johansen and Colella （1998, J. Comput. Phys. 147, 60） has been extended for elliptic and parabolic problems with interior boundaries or interfaces of discontinuities of material properties or solutions. Second order accuracy is achieved in space and time for both stationary and moving interface problems. The method is conservative for elliptic and parabolic problems with fixed interfaces. Based on this method, a front tracking algorithm for the Stefan problem has been developed. The accuracy of the method is measured through comparison with exact solution to a two-dimensional Stefan problem. The algorithm has been used for the study of melting and solidification problems.

An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications

A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces.Since the solution has low regularity across the interface,when applying finite difference discretization to this problem,an additional treatment accounting for the jump discontinuities must be employed.Here,we aim to elevate such an extra effort to ease our implementation by machine learning methodology.The key idea is to decompose the solution into singular and regular parts.The neural network learning machinery incorporating the given jump conditions finds the singular solution,while the standard five-point Laplacian discretization is used to obtain the regular solution with associated boundary conditions.Regardless of the interface geometry,these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation,making the hybrid method easy to implement and efficient.The two-and three-dimensional numerical results show that the present hybrid method preserves second-order accuracy for the solution and its derivatives,and it is comparable with the traditional immersed interface method in the literature.As an application,we solve the Stokes equations with singular forces to demonstrate the robustness of the present method.

Three Dimensional Interface Problems for Elliptic Equations

The author studies the structure of solutions to the interface problems for second order linear elliptic partial differential equations in three space dimension. The set of singular points consists of some singular lines and some isolated singular points. It is proved that near a singular line or a singular point, each weak solution can be decomposed into two parts, a singular part and a regular part. The singular parts are some finite sum of particular solutions to some simpler equations, and the regular parts are bounded in some norms, which are slightly weaker than that in the Sobolev space H^2.

A Fast Cartesian Grid-Based Integral Equation Method for Unbounded Interface Problems with Non-Homogeneous Source Terms

This work presents a fast Cartesian grid-based integral equation method for unbounded interface problems with non-homogeneous source terms.The unbounded interface problem is solved with boundary integral equation methods such that infinite boundary conditions are satisfied naturally.This work overcomes two difficulties.The first difficulty is the evaluation of singular integrals.Boundary and volume integrals are transformed into equivalent but much simpler bounded interface problems on rectangular domains,which are solved with FFT-based finite difference solvers.The second one is the expensive computational cost for volume integrals.Despite the use of efficient interface problem solvers,the evaluation for volume integrals is still expensive due to the evaluation of boundary conditions for the simple interface problem.The problem is alleviated by introducing an auxiliary circle as a bridge to indirectly evaluate boundary conditions.Since solving boundary integral equations on a circular boundary is so accurate,one only needs to select a fixed number of points for the discretization of the circle to reduce the computational cost.Numerical examples are presented to demonstrate the efficiency and the second-order accuracy of the proposed numerical method.

ANALYSIS OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR PARABOLIC INTERFACE PROBLEMS WITH NONSMOOTH INITIAL DATA

This article concerns numerical approximation of a parabolic interface problem with general L 2 initial value.The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface,with piecewise linear approximation to the interface.The semi-discrete finite element problem is furthermore discretized in time by the k-step backward difference formula with k=1,...,6.To maintain high-order convergence in time for possibly nonsmooth L 2 initial value,we modify the standard backward difference formula at the first k−1 time levels by using a method recently developed for fractional evolution equations.An error bound of O(t−k nτk+t−1 n h 2|log h|)is established for the fully discrete finite element method for general L 2 initial data.