Solving Interface Problems of the Helmholtz Equation by Immersed Finite Element Methods
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Abstract
This article reports our explorations for solving interface problems of the Helmholtz equation by immersed finite elements (IFE) on interface independent meshes. Two IFE methods are investigated: the partially penalized IFE (PPIFE) and discontinuous Galerkin IFE (DGIFE) methods. Optimal convergence rates are observed for these IFE methods once the mesh size is smaller than the optimal mesh size which is mainly dictated by the wave number. Numerical experiments also suggest that higher degree IFE methods are advantageous because of their larger optimal mesh size and higher convergence rates.
Keywords
Helmholtz interface problems Immersed finite element (IFE) methods Higher degree finite element methodsMathematics Subject Classification
65N30 65N50 35R051 Introduction
The Helmholtz boundary value problems (BVPs) without interface have been widely studied using numerical methods, among which, are the classic finite element methods [6, 8, 21, 33]. More sophisticated finite element methods such as the interior penalty Galerkin (IPG) method [14, 22] and discontinuous Galerkin (DG) method (including interior penalty DG, IPDG for abbreviation) [23, 24, 25, 38, 46, 47] have been developed for solving Helmholtz boundary value problems.
Finite element methods such as those mentioned above can be applied to solve interface problems provided that they use a bodyfitting mesh [7, 12, 17, 53] in which each element is essentially on one side of the interface or each element is occupied by one of the materials. However, this bodyfitting requirement can hinder a finite element method in some applications such as a simulation in which an interface problem has to be solved repeatedly for different geometries and locations of the interface which demand the mesh to be regenerated again and again to accommodate the change of the material interface. For such simulations, numerical methods based an interfaceindependent mesh are often preferred. For example, extended finite element methods (XFEM), immersed interface methods (IIM), and multiscale methods have been developed to solve interface problems of elliptic partial differential equations with an interfaceindependent mesh, and readers are referred to [5, 9, 19, 32, 39, 40] for more details about these methods. In particular, some of these methods have been applied to Helmholtz interface problems, see [55] for IIM and [57] for XFEM.
The immersed finite element (IFE) methods are another class of finite element methods developed to solve interface problems with interfaceindependent meshes. With HesieClough–Tocher type [11, 20] macro polynomials constructed according to the interface jump conditions as the local shape functions, IFE methods allow the interface to split the interior of the elements in a mesh; therefore, IFE methods are interfaceindependent methods that can use highly structured Cartesian meshes for problems with nontrivial interfaces, see [41, 42, 51] for some IFE spaces based on triangular Cartesian meshes and [26, 29, 31, 43] for some IFE spaces based rectangular Catesian meshes. Partially penalized IFE (PPIFE) and DGIFE methods [27, 30, 31, 44, 45, 56] have been developed for elliptic interface problems, and the related extensions to higher degree IFE methods are presented in [1, 2, 3, 48].
The purpose of this article is to report our exploration for applying IFE methods to the Helmholtz interface problem described by (1a)–(1d). Since the Helmholtz equation contains the elliptic operator and the interface jump conditions specified in (1c) and (1d) are the same as those for the elliptic interface problems, it is natural for us to consider IFE methods for the Helmholtz interface problem by using the IFE spaces constructed for the elliptic interface problems. Both the partially penalized Galerkin formulation and the discontinuous Galerkin formulation will be considered. We will also explore higher degree IFE methods because it is well known that higher degree finite element methods have desirable features for wave propagation problems [4, 15, 49], such as reducing the numerical dispersion and errors in solution due to the pollution effect caused by a large wave number [34, 52]. Additionally, it was found out that employing higher degree finite elements requires less degrees of freedom for numerical solutions to attain a specific accuracy [34]. Specifically, the layout of this article is as follows: in Sect. 2, we introduce the notation and assumptions to be used in this article, and we recall the IFE spaces in the literature. In Sect. 3, PPIFE and DGIFE schemes are derived for the Helmholtz interface problem (1). In Sect. 4, we present numerical examples to show the features of the proposed IFE methods. Some concise conclusions and remarks are given in Sect. 5.
2 IFE Spaces
 (H1)

The interface \(\Gamma \) cannot intersect an edge of any element at more than two points unless the edge is part of \(\Gamma \).
 (H2)

If \(\Gamma \) intersects the boundary of an element at two points, these intersection points must be on different edges of this element.
 (H3)

The interface \(\Gamma \) is a piecewise \(C^2\) function, and for every interface element \(T\in \mathcal {T}^\text{i}_h\), \(\Gamma \cap T\) is \(C^2\).
2.1 Local Linear and Bilinear IFE Spaces
2.2 Higher Degree IFE Spaces
3 IFE Methods for Helmholtz Interface Problems
Basically, all the IFE methods developed for the elliptic interface problems such as those in [1, 27, 41, 42, 44, 45] can be extended to Helmholtz interface problems. We will focus on the PPIFE method and the DGIFE method because it has been observed that, for the elliptic interface problems, the penalty terms in these methods enhance the stability as well as the accuracy.
3.1 PPIFE Methods for Helmholtz Interface Problems
Remark 3.1
The continuity of \(v_hT,~\forall T \in \mathcal {T}_h\) for every \(v_h \in S_h^1(\Omega )\) implies that the last three terms in the bilinear form \(a_h^{PP}(\cdot , \cdot )\) defined in (22) can be ignored in the PPIFE method based on the linear and bilinear IFE spaces.
3.2 DGIFE Method for Helmholtz Interface Problems
Remark 3.2
Similar to the PPIFE method, the continuity of \(v_hT,~\forall T \in \mathcal {T}_h\) for every \(v_h \in DS_h^1(\Omega )\) implies that the last three terms in the bilinear form \(a_h^{\text{DG}}(\cdot , \cdot )\) defined in (26) can be ignored in the DGIFE method based on the linear and bilinear IFE spaces.
4 Numerical Examples
For the PPIFE and DGIFE scheme, we choose \(\epsilon =1\), so the PPIFE and DGIFE schemes are symmetric PPIFE (SPPIFE)/symmetric DGIFE (SDGIFE) schemes, respectively, and we choose the penalty parameter \(\sigma _e^0=30 \mathrm {max}\{\beta _1, \beta _2\}\). We apply these IFE methods to Helmholtz interface problems with \(\beta _1=1\), \(\beta _2=5\) or 50 representing small and moderately large discontinuity in the coefficient \(\beta \), and to Helmholtz interface problems with \(w=10\) or 50 for small and larger wave numbers.
From Figs. 2, 3, 4 and 5, we also observe that the solution u oscillates more for the Helmholtz interface problem with a larger wave number, and u has a smaller magnitude in the subdomain where \(\beta \) has a larger value. It is well known that the oscillation in the exact solution u in the Helmholtz equation dictates the mesh size for its numerical solution; otherwise, the accuracy or convergence of the numerical solution cannot be guaranteed if the mesh size is not sufficiently small to resolve the oscillation. For the convergence, we recall the critical mesh size discussed in [24] that indicates when the numerical solutions start to converge:
Definition 4.1

(C1) \(e(h,w)<1\) for \(h<H(w,f)\),

(C2) \(e(h,w)\rightarrow 0\) as \(h\rightarrow 0\), where \(e(h,w)={u  u_h_{1,\Omega }}/{u_{1,\Omega }}\) is the relative error in the semi\(H^1\) norm.
By the data in Table 1, we can see that the linear PPIFE solution converges from a coarse mesh with \(h = 2/10\) when the wave number in the interface problem is small \(w = 10\), and this suggests the critical mesh size \(H(10, f) \lessapprox 2/10\). However, for the interface problem with a large wave number \(w = 50\) but a small discontinuity in \(\beta \), the critical mesh size for the PPIFE solution seems to be much smaller \(H(50, f) \lessapprox 2/170\). When the discontinuity in \(\beta \) is larger, the critical mesh size for the linear IFE solution \(H(50, f) \lessapprox 2/230\) which is even smaller. Even though not presented here for the sake of controlling the page consumption, we have observed in our numerical experiments that, for either small or large discontinuity in \(\beta \), the critical mesh size for the cubic PPIFE solution is about 2/10 when the wave number is small \(w =10\), but for a large wave number \(w = 50\) it becomes about 2/30. Therefore, higher degree IFE methods can start to converge on rather coarse mesh even for higher wave numbers. Similar behaviors are also observed for the DGIFE solutions.
From the data in Table 1, we can see that IFE solutions do not converge optimally until the mesh is further reduced beyond the critical mesh size, and this motivates us to introduce the optimal mesh size to characterize this phenomenon:
Definition 4.2

(O1) \(\Vert u  u_h\Vert _{0,\Omega }\approx Ch^{k+1}\) for \(h < {\tilde{H}}(w,f)\),

(O2) \(\vert u  u_h\vert _{1,\Omega }\approx Ch^{k}\) for \(h < {\tilde{H}}(w,f)\),
From the data presented in Tables 1 and 2, we can see that when the wave number is small \(w = 10\) and the discontinuity in \(\beta \) is small, the optimal mesh size for the linear PPIFE solution seems to be \({\tilde{H}}(10,f) \lessapprox 2/360\). But when the discontinuity in \(\beta \) is larger, the optimal mesh size for the linear PPIFE solution seems to be a little smaller \({\tilde{H}}(10,f) \lessapprox 2/420\). However, for a larger number \(w = 50\), the data in Table 1 indicate that the optimal mesh size for the linear PPIFE solution seems to be drastically smaller such that \({\tilde{H}}(50, f) \lessapprox 2/1 280\), and we note that \({\tilde{H}}(50, f) \ll H(50, f)\), i.e., the optimal mesh size for the linear PPIFE solution is much smaller than its critical mesh size. Similar characteristics is observed for bilinear PPIFE solution, linear DGIFE solution, and bilinear DGIFE solution, and this strongly demonstrates the inefficiency of using the lower degree method to solve a Helmholtz interface problem with a large wave number.
On the other hand, for this Helmholtz interface problem with a small wave number \(w = 10\), the data in Table 3 show that the optimal mesh size for the cubic PPIFE solution seems to be such that \({\tilde{H}}(10, f) \lessapprox 2/40\) when \(\beta \) has a small discontinuity, and \({\tilde{H}}(10, f) \lessapprox 2/80\) when \(\beta \) has a larger discontinuity. For a large wave number \(w = 50\), the data in Table 3 show that the optimal mesh size for the cubic PPIFE solution seems to be such that \({\tilde{H}}(50, f) \lessapprox 2/260\) which is much larger than the optimal mesh size for the linear PPIFE solution. Therefore, the higher degree IFE methods are advantageous because they can converge optimally on much coarser mesh than lower degree PPIFE methods. Similar behavior has also been observed for DGIFE methods in our numerical experiments.
We can also see the advantage of higher degree IFE methods from the point of view of the global degrees of freedom and accuracy. According to the discussions above, for a small wave number \(w = 10\) and a small discontinuity in \(\beta \), the linear PPIFE solution starts to converge optimally once its mesh size is such that \(h = {{{\tilde{H}}}}(10, f) \lessapprox 2/360\), and on such a mesh, the global degrees of freedom (GDOF) in this linear PPIFE solution is about \((361)^2 = 130321\). In comparison, the cubic PPIFE solution starts to converge optimally once its mesh size is such that \(h = {{{\tilde{H}}}}(10, f) \lessapprox 2/40\), and the GDOF in this cubic PPIFE solution is about \((3\times 40 + 1)^2 = 14641\) which is about 9 times smaller than the GDOF of the linear PPIFE solution. Far more importantly, by comparing data in Tables 1, 2, and 3, we can see that, on meshes whose mesh sizes are smaller than the optimal mesh sizes, the cubic PPIFE solution is obviously far more accurate than the linear PPIFE solution even though the GDOF of the linear PPIFE solution is much larger. Similar advantages are also observed for higher degree DGIFE methods. Therefore, for wave propagation interface problems, the higher degree IFE methods should be preferred even though the development of higher degree IFE methods is still in its early stage, and its research deserves more attention.
By design, the linear and bilinear IFE spaces are consistent with their corresponding FE spaces, i.e., the linear/bilinear IFE space becomes linear/bilinear FE space when \(\beta _1 = \beta _2\). Additionally, the formulations for the PPIFE method and the FE method are quite close to each other, and they differ only on interface elements whose union forms a small band around the interface. Therefore, it is interesting to know how the PPIFE and FE solutions behave from the point of views of the critical mesh size and the optimal mesh size. From the data in Table 4, when the wave number is small \(w = 10\), the critical mesh size for the linear FE solution is about 2 / 10 which is not much different from the critical mesh size for the linear IFE solution according to the data in Table 1. For a larger wave number \(w = 50\), the data in Table 4 indicate that the critical mesh size for the linear FE solution is about 2 / 140 which is again not much different from the critical mesh size for the linear IFE solution which is about 2 / 170 when the discontinuity in \(\beta \) is small according to the data in Table 1, but the difference becomes a little more obvious when the discontinuity is larger. We also have observed that the critical mesh size of a cubic IFE method is just slightly smaller than that of its FE counterpart. As for the optimal mesh size, by Table 4, when the wave number is small \(w = 10\), the optimal mesh size for the linear FE solution seems to be about 2 / 100 which is obviously larger than the optimal mesh for the linear PPIFE solution which is about 2 / 360 when the discontinuity in \(\beta \) is small. However, for a large wave number \(w = 50\), the data in Table 5 suggest that the optimal mesh for the cubic FE solution is about 2 / 260 which is comparable to the optimal mesh size for the cubic PPIFE solution suggested by the data in Table 3. Similar behaviors are also observed for DGIFE and DG methods. In summary, for Helmholtz problems, higher degree IFE methods and FE methods behave somewhat similarly, especially when the discontinuity in \(\beta \) is small or from the point of view of the optimal mesh size.
5 Conclusions
Errors in linear SPPIFE solution and convergence rates for \(\beta _1=1\), different \(\beta _2\) and w
w  \(\beta _2\)  N  \(\Vert u  u_h\Vert _{0,\Omega }\)  Rate  \(u  u_h_{1,\Omega }\)  Rate  e(h, w) 

10  5  10  8.2104E−02  NA  7.4639E−01  NA  7.2748E−01 
20  4.9816E−02  0.7208  4.7736E−01  0.6448  4.6526E−01  
40  1.6991E−02  1.5519  2.0544E−01  1.2164  2.0023E−01  
80  4.5312E−03  1.9068  8.7699E−02  1.2281  8.5477E−02  
160  1.1495E−03  1.9789  4.1293E−02  1.0867  4.0247E−02  
320  2.8953E−04  1.9892  2.0298E−02  1.0245  1.9784E−02  
640  7.2356E−05  2.0005  1.0106E−02  1.0062  9.8499E−03  
1280  1.8062E−05  2.0021  5.0476E−03  1.0015  4.9197E−03  
10  50  10  6.4644E−02  NA  7.5030E−01  NA  7.5659E−01 
20  3.3762E−02  0.9371  4.0855E−01  0.8769  4.1197E−01  
40  2.0716E−02  0.7047  2.4551E−01  0.7347  2.4756E−01  
80  1.0567E−02  0.9712  1.2749E−01  0.9454  1.2856E−01  
160  3.4762E−03  1.6039  5.1411E−02  1.3102  5.1841E−02  
320  9.1923E−04  1.9190  2.1448E−02  1.2612  2.1627E−02  
640  2.3301E−04  1.9800  1.0022E−02  1.0976  1.0106E−02  
1280  5.8619E−05  1.9910  4.9165E−03  1.0275  4.9576E−03  
50  5  80  3.9451E−02  − 0.3967  1.6817E+00  − 0.3401  1.4876E+00 
160  2.8478E−02  0.4702  1.1843E+00  0.5059  1.0357E+00  
170  2.5688E−02  1.7007  1.0693E+00  1.6846  9.4531E−01  
180  2.2661E−02  2.1937  9.4548E−01  2.1536  8.3582E−01  
190  1.9804E−02  2.4923  8.2939E−01  2.4231  7.3319E−01  
200  1.7282E−02  2.6557  7.2717E−01  2.5642  6.4283E−01  
320  5.1669E−03  2.5689  2.3826E−01  2.3740  2.1095E−01  
640  1.2047E−03  2.1006  7.5049E−02  1.6666  6.6484E−02  
1280  2.7773E−04  2.1170  3.0119E−02  1.3171  2.6673E−02  
2560  6.8813E−05  2.0129  1.4235E−02  1.0813  1.2664E−02  
50  50  160  4.0513E−02  0.0049  1.8752E+00  0.0178  1.7055E+00 
210  3.3354E−02  0.7150  1.5637E+00  0.6680  1.4222E+00  
220  2.7486E−02  4.1599  1.2931E+00  4.0842  1.1761E+00  
230  2.2856E−02  4.1500  1.0792E+00  4.0680  9.8156E−01  
240  1.9322E−02  3.9462  9.1595E−01  3.8543  8.3306E−01  
250  1.6556E−02  3.7845  7.8804E−01  3.6846  7.1673E−01  
320  7.5321E−03  3.1904  3.7059E−01  3.0562  3.3705E−01  
640  1.4355E−03  2.3915  8.6787E−02  2.0943  7.8931E−02  
1280  3.3874E−04  2.0833  3.1457E−02  1.4641  2.8610E−02  
2560  8.3515E−05  2.0201  1.4101E−02  1.1576  1.2825E−02 
Errors in linear SPPIFE solution and convergence rates for \(\beta _1=1\), different \(\beta _2\) and w
w  \(\beta _2\)  N  \(\Vert u  u_h\Vert _{0,\Omega }\)  Rate  \(u  u_h_{1,\Omega }\)  Rate  e(h, w) 

10  5  320  2.8953E−04  NA  2.0298E−02  NA  1.9784E−02 
330  2.7060E−04  2.1969  1.9676E−02  1.0115  1.9178E−02  
340  2.5617E−04  1.8364  1.9092E−02  1.0099  1.8608E−02  
350  2.4053E−04  2.1725  1.8540E−02  1.0115  1.8070E−02  
360  2.2829E−04  1.8539  1.8022E−02  1.0064  1.7565E−02  
370  2.1555E−04  2.0964  1.7529E−02  1.0129  1.7084E−02  
380  2.0469E−04  1.9384  1.7066E−02  1.0037  1.6633E−02  
390  1.9430E−04  2.0050  1.6622E−02  1.0129  1.6201E−02  
400  2.0469E−04  2.0265  1.6206E−02  1.0029  1.5795E−02  
410  1.9430E−04  1.9657  1.5806E−02  1.0114  1.5405E−02  
10  50  400  5.9551E−04  NA  1.5353E−02  NA  1.6780E−02 
420  5.4036E−04  1.9919  1.5761E−02  1.1128  1.5893E−02  
440  4.9144E−04  2.0400  1.4969E−02  1.1085  1.5094E−02  
460  4.5041E−04  1.9613  1.4260E−02  1.0924  1.4379E−02  
480  4.1440E−04  1.9577  1.4260E−02  1.0840  1.3731E−02  
500  3.8156E−04  2.0225  1.3617E−02  1.0848  1.3136E−02  
520  3.5217E−04  2.0434  1.3027E−02  1.0795  1.2591E−02  
540  3.2554E−04  2.0840  1.1996E−02  1.0773  1.2090E−02  
560  2.8306E−04  1.9568  1.1538E−02  1.0611  1.1207E−02  
580  2.6451E−04  1.9994  1.1118E−02  1.0574  1.0812E−02 
Errors in cubic SPPIFE solution and convergence rates for \(\beta _1=1\), different \(\beta _2\) and w
w  \(\beta _2\)  N  \(\Vert u  u_h\Vert _{0,\Omega }\)  Rate  \(u  u_h_{1,\Omega }\)  Rate  e(h, w) 

10  5  40  1.7107E−06  NA  4.1486E−04  NA  4.1486E−04 
50  6.9536E−07  4.0343  2.1274E−04  2.9930  2.1274E−04  
60  3.3424E−07  4.0180  1.2326E−04  2.9937  1.2326E−04  
70  1.7995E−07  4.0168  7.7668E−05  2.9960  7.7668E−05  
80  1.0543E−07  4.0033  5.2073E−05  2.9941  5.2073E−05  
10  50  80  1.0522E−07  NA  5.1511E−05  NA  5.1942E−05 
90  6.5444E−08  4.0319  3.6176E−05  3.0005  3.6478E−05  
100  4.2864E−08  4.0164  2.6386E−05  2.9950  2.6607E−05  
110  2.9224E−08  4.0187  1.9829E−05  2.9973  1.9995E−05  
120  2.0628E−08  4.0033  1.5276E−05  2.9982  1.5404E−05  
130  1.4951E−08  4.0218  1.2009E−05  3.0060  1.2110E−05  
50  5  160  1.0869E−06  4.3450  9.8166E−04  2.9970  8.6781E−04 
180  6.5893E−07  4.2427  6.8980E−04  2.9983  6.0979E−04  
200  4.2458E−07  4.1736  5.0307E−04  2.9978  4.4472E−04  
220  2.8662E−07  4.1252  3.7808E−04  2.9972  3.3423E−04  
240  2.0079E−07  4.0882  2.9128E−04  2.9976  2.5750E−04  
260  1.4498E−07  4.0646  2.2913E−04  2.9993  2.0256E−04  
280  1.0736E−07  4.0497  1.8349E−04  3.0001  1.6220E−04  
300  8.1240E−08  4.0363  1.4921E−04  2.9984  1.3190E−04  
320  6.2625E−08  4.0245  1.2296E−04  2.9963  1.0870E−04  
50  50  160  1.0830E−06  4.4365  9.4940E−04  2.9972  8.6348E−04 
180  6.5049E−07  4.3017  6.6717E−04  2.9970  6.0680E−04  
200  4.1663E−07  4.2111  4.8660E−04  2.9963  4.4257E−04  
220  2.8016E−07  4.1513  3.6573E−04  2.9960  3.3263E−04  
240  1.9577E−07  4.1099  2.8179E−04  2.9962  2.5629E−04  
260  1.4112E−07  4.0807  2.2170E−04  2.9956  2.0164E−04  
280  1.0438E−07  4.0646  1.7754E−04  2.9965  1.6147E−04  
300  7.8916E−08  4.0491  1.4437E−04  2.9958  1.3131E−04  
320  6.0794E−08  4.0364  1.1898E−04  2.9948  1.0821E−04 
Errors in linear FE solution for \(\beta _1 = \beta _2 = 1\)
w  \(\beta _2\)  N  \(\Vert u  u_h\Vert _{0,\Omega }\)  Rate  \(u  u_h_{1,\Omega }\)  Rate  e(h, w) 

10  1  10  1.4494E−01  NA  1.5230E+00  NA  9.2162E−01 
20  6.4454E−02  1.1692  7.9105E−01  0.9450  4.7870E−01  
40  1.8912E−02  1.8572  3.1627E−01  1.3221  1.9139E−01  
80  4.9128E−03  1.9713  1.3738E−01  1.1335  8.3135E−02  
90  3.8919E−03  1.9778  1.2056E−01  1.1089  7.2956E−02  
100  3.1583E−03  1.9823  1.0748E−01  1.0901  6.5040E−02  
110  2.6138E−03  1.9856  9.7007E−02  1.0755  5.8703E−02  
50  1  80  4.2659E−02  NA  2.1287E+00  NA  1.2300E+00 
90  4.1340E−02  0.2668  2.0649E+00  0.2584  1.1931E+00  
100  4.0504E−02  0.1937  2.0246E+00  0.1874  1.1698E+00  
110  3.9731E−02  0.2023  1.9877E+00  0.1929  1.1485E+00  
120  3.8340E−02  0.4094  1.9218E+00  0.3876  1.1104E+00  
130  3.6248E−02  0.7011  1.8218E+00  0.6671  1.0527E+00  
140  3.3800E−02  0.9438  1.7040E+00  0.9021  9.8458E−01  
150  3.1281E−02  1.1224  1.5821E+00  1.0758  9.1415E−01  
160  2.8836E−02  1.2611  1.4632E+00  1.2106  8.4544E−01 
Errors in cubic FE solution for \(\beta _1= \beta _2 = 1\)
w  \(\beta _2\)  N  \(\Vert u  u_h\Vert _{0,\Omega }\)  Rate  \(u  u_h_{1,\Omega }\)  Rate  e(h, w) 

50  1  240  3.1879E−07  4.1290  4.6238E−04  2.9985  2.6717E−04 
250  2.6955E−07  4.1100  4.0911E−04  2.9986  2.3638E−04  
260  2.2952E−07  4.0982  3.6372E−04  2.9987  2.1016E−04  
270  1.9674E−07  4.0840  3.2480E−04  2.9987  1.8767E−04  
280  1.6963E−0  4.0763  2.9124E−04  2.9989  1.6828E−04  
290  1.4708E−07  4.0655  2.6215E−04  2.9989  1.5147E−04  
300  1.2816E−07  4.0607  2.3680E−04  2.9990  1.3683E−04  
310  1.1222E−07  4.0518  2.1463E−04  2.9990  1.2401E−04  
320  9.8681E−08  4.0491  1.9513E−04  2.9991  1.1275E−04 
Notes
Acknowledgements
This research was partially supported by GRF BQ56D of HKSAR and Polyu GUA7V.
References
 1.Adjerid, S., BenRomdhane, M., Lin, T.: Higher degree immersed finite element methods for secondorder elliptic interface problems. Int. J. Numer. Anal. Model. 11(3), 541–566 (2014)MathSciNetzbMATHGoogle Scholar
 2.Adjerid, S., BenRomdhane, M., Lin, T.: Higher degree immersed finite element spaces constructed according to the actual interface. Comput. Math. Appl. 75(6), 1868–1881 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
 3.Adjerid, S., Guo, R., Lin, T.: High degree immersed finite element spaces by a least squares method. Int. J. Numer. Anal. Model. 14(4/5), 604–626 (2017)MathSciNetzbMATHGoogle Scholar
 4.Ainsworth, M.: Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198, 106–130 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
 5.Annavarapu, C., Hautefeuille, M., Dolbow, J.: A finite element method for crack growth without remeshing. Comput. Methods Appl. Mech. Eng. 225–228, 44–54 (2012)zbMATHCrossRefGoogle Scholar
 6.Aziz, A.K., Werschulz, A.: On the numerical solutions of Helmholtz’s equation by the finite element method. SIAM J. Numer. Anal. 19(5), 166–178 (1995)MathSciNetGoogle Scholar
 7.Babuška, I.: The finite element method for elliptic equations with discontinuous coefficients. Computing (Arch. Elektron. Rechnen) 5, 207–213 (1970)MathSciNetzbMATHGoogle Scholar
 8.Babuška, I.M., Sauter, S.A.: Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? Comput. Math. Appl. 34(6), 2392–2423 (1997)MathSciNetzbMATHGoogle Scholar
 9.Barrett, J.W., Elliott, C.M.: Fitted and unfitted finiteelement methods for elliptic equations with smooth interfaces. IMA J. Numer. Anal. 7(3), 283–300 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
 10.BonnetBen Dhia, A.S., Ciarlet Jr., P., Zwölf, C.M.: Time harmonic wave diffraction problems in materials with signshifting coefficients. J. Comput. Appl. Math. 234, 1912–1919 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
 11.Braess, D.: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 2nd edn. Cambridge University Press, Cambridge (2001). Translated from the 1992 German edition by Larry L. SchumakerzbMATHGoogle Scholar
 12.Bramble, J.H., King, J.T.: A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math. 6(2), 109–138 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
 13.Brown, D.L.: A note on the numerical solution of the wave equation with piecewise smooth coefficients. Math. Comput. 42(166), 369–391 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
 14.Burman, E., Wu, H., Zhu, L.: Linear continuous interior penalty finite element method for Helmholtz equation with high wave number: onedimensional analysis. Numer. Methods Partial Differ. Equ. 32(5), 1378–1410 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
 15.Bériot, H., Prinn, A., Gabard, G.: Efficient implementation of highorder finite elements for Helmholtz problems. Int. J. Numer. Methods Eng. 106, 213–240 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
 16.Chandlerwilde, S.N., Zhang, B.: Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers. SIAM J. Math. Anal. 30(3), 559–583 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
 17.Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79(2), 175–202 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
 18.Christiansen, P.S., Krenk, S.: A recursive finite element technique for acoustic fields in pipes with absorption. J. Sound Vib. 122(1), 107–118 (1988)CrossRefGoogle Scholar
 19.Chu, C.C., Graham, I.G., Hou, T.Y.: A new multiscale finite element method for highcontrast elliptic interface problems. Math. Comput. 79(272), 1915–1955 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
 20.Clough, R.W., Tocher, J.L.: Finite element stiffness matrices for analysis of plate bending. In: Przemieniecki, J.S., Bader,R.M., Bozich, W.F., Johnson, J.R., Mykytow, W.J. (eds.) Matrix Methods in Structural Mechanics, The Proceedings of the Conference held at WrightParrterson Air Force Base, Ohio, 26–28, October, 1965, pp. 515–545, Washington, 1966. Air Force Flight Dynamics LaboratoryGoogle Scholar
 21.Douglas Jr., J., Sheen, D., Santos, J.E.: Approximation of scalar waves in the spacefrequency domain. Math. Models Methods Appl. Sci. 4(4), 509–531 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
 22.Du, Y., Wu, H.: Preasymptotic error analysis of higher order FEM and CIPFEM for Helmholtz equation with high wave number. SIAM J. Numer. Anal. 53(2), 782–804 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
 23.Farhat, C., Harari, I., Hetmaniuk, U.: A discontinuous Galerkin method with lagrange multipliers for the solution of Helmholtz problems in the midfrequency regime. Comput. Methods Appl. Mech. Eng. 192, 1389–1419 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
 24.Feng, X., Wu, H.: Discontinuous Galerkin methods for the Helmholtz equation with large wave number. SIAM J. Numer. Anal. 47(4), 2872–2896 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
 25.Gittelson, G., Hiptmair, R., Perugia, I.: Plane wave discontinuous Galerkin methods: analysis of the hversion. Esaim Math. Model. Numer. Anal. 43, 297–331 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
 26.He, X., Lin, T., Lin, Y.: Approximation capability of a bilinear immersed finite element space. Numer. Methods Partial Differ. Equ. 24(5), 1265–1300 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
 27.He, X., Lin, T., Lin, Y.: Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient. J. Syst. Sci. Complex. 23(3), 467–483 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
 28.He, X., Lin, T., Lin, Y.: Immersed finite element methods for elliptic interface problems with nonhomogeneous jump conditions. Int. J. Numer. Anal. Model. 8(2), 284–301 (2011)MathSciNetzbMATHGoogle Scholar
 29.He, X., Lin, T., Lin, Y.: The convergence of the bilinear and linear immersed finite element solutions to interface problems. Numer. Methods Partial Differ. Equ. 28(1), 312–330 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
 30.He, X., Lin, T., Lin, Y.: A selective immersed discontinuous Galerkin method for elliptic interface problems. Math. Methods Appl. Sci. 37(7), 983–1002 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
 31.He, X.: Bilinear immersed finite elements for interface problems. PhD thesis, Virginia Polytechnic Institute and State University (2009)Google Scholar
 32.Hou, T.Y., Wu, X.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
 33.Ihlenburg, F., Babuška, I.: Finite element solution of the Helmholtz equation with high wave number. I. The hversion of the FEM. Comput. Math. Appl. 30(9), 9–37 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
 34.Ihlenburg, F., Babuška, I.: Finite element solutions of the Helmholtz equation with high wave number part II: the hp version of the FEM. SIAM J. Numer. Anal. 34(1), 315–358 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
 35.Jensen, F.B., Kuperman, W.A., Porter, M.B., Schmidt, H.: Computational Ocean Acoustics. Springer, Berlin (1995)zbMATHGoogle Scholar
 36.Klenow, B., Nisewonger, A., Batra, R.C., Brown, A.: Reflection and transmission of plane waves at an interface between two fluids. Comput. Fluids 36, 1298–1306 (2007)zbMATHCrossRefGoogle Scholar
 37.Kreiss, H., Petersson, N.A.: An embedded boundary method for the wave equation with discontinuous coefficients. SIAM J. Sci. Comput. 28(6), 2054–2074 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
 38.Lam, C.Y., Shu, C.W.: A phasebased interior penalty discontinuous Galerkin method for the Helmholtz equation with spatially varying wavenumber. Comput. Methods Appl. Mech. Eng. 318, 456–473 (2017)MathSciNetCrossRefGoogle Scholar
 39.LeVeque, R.J., Li, Z.L.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31(4), 1019–1044 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
 40.Li, Z., Ito, K.: The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains. Frontiers in Applied Mathematics, vol. 33. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006)zbMATHCrossRefGoogle Scholar
 41.Li, Z., Lin, T., Lin, Y., Rogers, R.C.: An immersed finite element space and its approximation capability. Numer. Methods Partial Differ. Equ. 20(3), 338–367 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
 42.Li, Z., Lin, T., Wu, X.: New Cartesian grid methods for interface problems using the finite element formulation. Numer. Math. 96(1), 61–98 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
 43.Lin, T., Lin, Y., Rogers, R., Lynne Ryan, M.: A Rectangular Immersed Finite Element Space for Interface Problems. Scientific Computing and Applications (Kananaskis, AB, 2000). Advances in Computation: Theory and Practice, vol. 7, pp. 107–114. Nova Sci. Publ, Huntington (2001)zbMATHGoogle Scholar
 44.Lin, T., Lin, Y., Zhang, X.: Partially penalized immersed finite element methods for elliptic interface problems. SIAM J. Numer. Anal. 53(2), 1121–1144 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
 45.Lin, T., Yang, Q., Zhang, X.: A priori error estimates for some discontinuous Galerkin immersed finite element methods. J. Sci. Comput. 65, 875–894 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
 46.Parsania, A., Melenk, J.M., Sauter, D.: General DGmethods for highly indefinite Helmholtz problems. J. Sci. Comput. 57, 536–581 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
 47.Perugia, I.: A note on the discontinuous Galerkin approximation of the Helmholtz equation. Lecture notes. ETH, Zürich (2006)Google Scholar
 48.Romdhane, M.B.: Higherdegree immersed finite elements for secondorder elliptic interface problems. PhD thesis, Virginia Polytechnic Institute and State University (2011)Google Scholar
 49.Semblat, J.F., Brioist, J.J.: Efficiency of higher order finite elements for the analysis of seismic wave propagation. J. Sound Vib. 231(2), 460–467 (2000)CrossRefGoogle Scholar
 50.Speck, F.O.: Sommerfeld diffraction problems with first and second kind boundary conditions. SIAM J. Math. Anal. 20(2), 396–407 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
 51.Suater, S.A., Warnke, R.: Composite finite elements for elliptic boudnary value problems with discontinuous coefficients. Computing 77, 29–55 (2006)MathSciNetCrossRefGoogle Scholar
 52.Wang, K., Wong, Y.: Pollutionfree finite difference schemes for nonhomogeneous Helmholtz equation. Int. J. Numer. Anal. Model. 11(4), 787–815 (2014)MathSciNetGoogle Scholar
 53.Xu, J.: Estimate of the convergence rate of the finite element solutions to elliptic equation of second order with discontinuous coefficients. Nat. Sci. J. Xiangtan Univ. 1, 1–5 (1982)Google Scholar
 54.Zhang, J.: Wave propagation across fluidsolid interfaces: a grid method approach. Geophys. J. Int. 159, 240–252 (2004)CrossRefGoogle Scholar
 55.Zhang, S.M., Li, Z.: An augmented IIM for Helmholtz/Poisson equations on irregular domains in complex space. Int. J. Numer. Anal. Model. 13(1), 166–178 (2016)MathSciNetzbMATHGoogle Scholar
 56.Zhang, X.: Nonconforming immersed finite element methods for interface problems. PhD thesis, Virginia Polytechnic Institute and State University (2013)Google Scholar
 57.Zou, Z., Aquino, W., Harari, I.: Nitsche's method for Helmholtz problems with embedded interfaces. Int. J. Numer. Meth. Eng. 110, 618–636 (2017)MathSciNetzbMATHCrossRefGoogle Scholar