Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces
Introduction
Since the pioneer work of Peskin [50] in 1977, much attention has been paid to the numerical solution of elliptic equations with discontinuous coefficients and singular sources on regular Cartesian grids [7], [8], [11], [15], [18], [20], [30], [31], [32], [55], [58]. Simple Cartesian grids are preferred in these studies since the complicated procedure of generating unstructured grid could be bypassed, and well developed fast algebraic solvers could be utilized. The importance of elliptic interface problems has been well recognized in a variety of disciplines, such as fluid dynamics [16], [19], [29], [47], electromagnetics [23], [24] and material science [26]. However, to construct highly efficient methods for these problems is a difficult task due to the low global regularity of the solution. Traditional numerical methods that are constructed with the assumption of smooth solutions cannot perform at designed accuracy, and might even diverge. For this class of problems, apart from Peskin’s immersed boundary method (IBM) [21], [33], [50], [51], [52], a number of other elegant methods have been proposed. Among them, the immersed interface method (IIM), proposed by LeVeque and Li [35] is a second order sharp interface scheme. The IIM has been made robust and efficient over the past decade [1], [14], [36], [37], [54]. The ghost fluid method (GFM) [17] was proposed as a relatively simple and easy to use approach. For irregular interfaces, it is nature to construct a solution in the finite element method formulation [2], [9], [38], in particular, using the discontinuous Galerkin technique [22]. A relevant, while quite distinct approach is the integral equation method for complex geometry [44], [45]. Aforementioned methods have found much success in scientific and engineering applications [6], [7], [8], [15], [18], [20], [25], [26], [27], [28], [30], [32], [34], [39], [41], [40], [42], [53], [54], [57], [58], [59]. A possible further direction in the field could be the development of higher order interface methods [20], [60], [61] which are particularly desirable for problems involving both material interfaces and high frequency oscillations, such as the interaction of turbulence and shock, and high frequency wave propagation in inhomogeneous media [5].
One of the most challenging problems in the field is the solution of elliptic equations with sharp-edged coefficients, i.e., non-smooth interfaces. Numerical solutions to this class of problems have widespread applications in science and engineering, such as electromagnetic wave scattering and propagation [12], [48], [49], wave-guides analysis [46], plasma–surface interaction [43], friction modeling [56] and turbulent-flow [4]. To the best of our knowledge, none of the aforementioned methods proposed for elliptic interface problems have been directly applied to the treatment of sharp-edged interfaces. Essentially, as the gradient near the tips of sharp-edged interface is not well defined, some earlier interface methods might not work. Most existing results on this class of problems are obtained by using finite element methods [46], [49]. However, finite element methods might exhibit a reduced convergence rate when used for the analysis of geometries containing sharp edges [25], [49]. Consequently, dramatic local mesh refinement is required in the vicinity of sharp edges [13], and leads to severe increase in computational time and memory requirement. In particular, local mesh refinement does not work if the solution is highly oscillatory due to the so-called pollution effect [3], which is a common situation in dealing with electromagnetic wave scattering and propagation. Hou and Liu proposed a finite element formulation [25] for solving elliptic equations with sharp-edged interfaces. Remarkably, these authors have achieved about 0.8th order convergence with non-body-fitting grids.
The objective of the present work is to extend the matched interface and boundary (MIB) method previously designed for solving elliptic problems with curved interfaces to problems with sharp-edged interfaces, thin-layered interfaces and interfaces that intersect with the geometric boundary. The MIB was proposed by Zhao and Wei [60] as a systematic higher-order method for electromagnetic wave propagation and scattering in dielectric media. Recently, it has been generalized for solving elliptic equations with curved interfaces by Zhou et al. [61]. The MIB approach makes use of fictitious domains so that the standard high order central finite difference (FD) method can be applied across the interface without the loss of accuracy. The fictitious values on fictitious domains are determined from enforcing the interface jump conditions at the exact position of the interface. One feature of the MIB is that it disassociates between the discretization of the elliptic equation and the enforcement of interface jump conditions. Another feature is to make repeated use of the lowest order jump conditions to determine the fictitious values on extended domains. Since only lowest order interface jump conditions are repeatedly used in the MIB method, arbitrarily high order convergence can be achieved in principle. For straight interfaces, MIB schemes of up to 16th order have been constructed [60], [61]. For lightly curved interfaces, up to 6th order schemes have been demonstrated [61]. Most recently, we have proposed an interpolation formulation of the MIB method without the explicit use of fictitious values [62]. We have shown that our interpolation formulation is equivalent to our earlier fictitious domain formulation. Fourth order convergence is obtained for arbitrarily curved interfaces. In the present work, we further generalize the MIB method to allow the presence of sharp-edged interfaces, thin-layered interfaces and interfaces that intersect with the domain boundary. For these problems, flexible strategies that have not been ever considered before are required. We introduce the concepts of primary and secondary fictitious values to overcome the difficulty of sharp-edged interfaces. The essence is to replace unavailable auxiliary points by secondary fictitious points to resolve primary fictitious values when there are geometric difficulties. The topological relations between the interfaces and the Cartesian mesh lines are classified into five distinct types. For each topology, appropriate secondary fictitious values, auxiliary points and jump conditions will be selected. In this work, the classification procedure and the solution scheme are made systematic and automatic. The generalized MIB method is designed to have 2nd order accuracy even if the interface is Lipschitz continuous while not C1. It captures sharp kinks at the interface without a priori knowledge of these kinks. The problem becomes more difficult as the interface edge is getting sharper. For an acute angle that is larger than the critical value , the present method can handle with the designed 2nd order convergence. However, edges with their acute angles smaller than the critical value can also be treated to the 2nd order accuracy if the mesh lines that bisect edges are not vertical or horizontal, which is true in most problems. For the problems with smooth (say C1) interfaces, the present method can easily achieve 2nd order convergence since required auxiliary points can always be found by refining the mesh.
It is well known that Galerkin formulations can directly solve problems with weak solution [10], [25]. Whereas, collocation formulations cannot directly handle this class of problems. Nevertheless, a standard technique is to multiply the solution with an appropriate polynomial factor [49] to remove the singularity. A new equation can be derived and solved. Then, it is matter of algebraic operation to resolve the original solution. This approach is incorporated in the present method to deal with problems with weak solutions.
The rest of this paper is organized as follows. In Section 2, we introduce the generalized MIB method that is able to treat sharp-edged interfaces, thin-layered interfaces and interfaces that intersect with geometric boundary. The critical angle of the sharp edge is analyzed. Both on-interface and off-interface schemes are proposed to handled five different topological relations. In Section 3, the proposed MIB method is validated for a wide variety of problems, ranging from those with critical small angles, edge tip on a grid point, highly oscillatory solution, multiple edges, multiply connected interfaces, a missile geometry, thin-layered coatings and a weak solution. This paper ends with a conclusion.
Section snippets
Theory and algorithm
In this section, we briefly describe the mathematical problem. Irregular points that are on the interface are treated differently from those off the interface. A pseudo-code is provided for the present method. We analyze the critical acute angle that can be treated by the proposed method.
Numerical studies
In this section, we examine the performance of the proposed MIB scheme for solving the Poisson equation with sharp-edged interfaces, thin-layered interfaces and interfaces that intersect with the geometric boundary. We consider six different interface geometries coupled with various boundary conditions and solution behaviors. In the first test case, we examine capability of the present MIB method in treating the critical sharp edge. We also test the present scheme for grid points exactly at the
Conclusion
A wide variety of scientific and engineering problems involve sharp-edged material interfaces, and their governing equations are of elliptic type. These problems call for new efficient methods that do not depend on massive local mesh refinement, which does not work for highly oscillatory waves due to the pollution effect [3]. The present work provides a solution to this class of problems on the Cartesian grid by extending the marched interface and boundary (MIB) method [60], [61] previously
Acknowledgments
This work was supported in part by NSF Grant IIS-0430987 and NSF Grant DMS-0616704.
References (62)
- et al.
Numerical-simulation of turbulent-flow over surface-mounted obstacles with sharp edges and corners
J. Wind Engng. Indust. Aerodyn.
(1990) A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions
J. Comput. Phys.
(2004)- et al.
A fast solver for the Stokes equations with distributed forces in complex geometries
J. Comput. Phys.
(2004) - et al.
An upwinding embedded boundary method for Maxwell’s equations in media with material interfaces: 2D case
J. Comput. Phys.
(2003) - et al.
Three-dimensional elliptic solvers for interface problems and applications
J. Comput. Phys.
(2003) - et al.
A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)
J. Comput. Phys.
(1999) - et al.
A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem
J. Comput. Phys.
(2005) - et al.
On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems
J. Comput. Phys.
(2005) High-order accurate methods in time-domain computational electromagnetics. A review
Adv. Imaging Electron Phys.
(2003)- et al.
A numerical method for solving variable coefficient elliptic equation with interfaces
J. Comput. Phys.
(2005)
A hybrid method for moving interface problems with application to the Hele–Shaw flow
J. Comput. Phys.
Reactive autophobic spreading of drops
J. Comput. Phys.
Robust numerical simulation of porosity evolution in chemical vapor infiltration II. Two-dimensional anisotropic fronts
J. Comput. Phys.
A Cartesian grid embedding boundary method for Poisson’s equation on irregular domains
J. Comput. Phys.
An immersed boundary method with formal second-order accuracy and reduced numerical viscosity
J. Comput. Phys.
A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains
J. Comput. Phys.
A fast Poisson solver for complex geometries
J. Comput. Phys.
Numerical analysis of blood flow in heart
J. Comput. Phys.
A 3-dimensional computational method for blood-flow in the heart. 1. Immersed elastic fibers in a viscous incompressible fluid
J. Comput. Phys.
Two-dimensional modelling of the river Rhine
J. Comput. Appl. Math.
Structural boundary design via level set and immersed interface methods
J. Comput. Phys.
Numerical approximations of singular source terms in differential equations
J. Comput. Phys.
High order FDTD methods via derivative matching for Maxwell’s equations with material interfaces
J. Comput. Phys.
High order matched interface and boundary (MIB) schemes for elliptic equations with discontinuous coefficients and singular sources
J. Comput. Phys.
On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method
J. Comput. Phys.
The immersed interface/multigrid methods for interface problems
SIAM J. Sci. Comput.
The finite element method for elliptic equations with discontinuous coefficients
Computing
Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave number?
SIAM J. Numer. Anal.
Numerical solution of the Helmholtz equation with high wave numbers
Int. J. Numer. Methods Engng.
Level set methods for geometric inverse problems in linear elasticity
Inverse Problems
A finite element method for interface problems in domains with smooth boundaries and interfaces
Adv. Comput. Math.
Cited by (164)
A Cartesian mesh approach to embedded interface problems using the virtual element method
2024, Journal of Computational PhysicsA new MIB-based time integration method for transient heat conduction analysis of discrete and continuous systems
2024, International Journal of Heat and Mass TransferSixth-order hybrid finite difference methods for elliptic interface problems with mixed boundary conditions
2024, Journal of Computational PhysicsPoisson-Boltzmann-based machine learning model for electrostatic analysis
2024, Biophysical JournalA correction function-based kernel-free boundary integral method for elliptic PDEs with implicitly defined interfaces
2024, Journal of Computational PhysicsTwo-dimensional orthotropic plate problems in a thermal environment: Refined crack modelling
2023, European Journal of Mechanics, A/Solids