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In this paper, we develop a new immersed finite element method (IFEM) for two-phase incompressible Stokes flows. We allow the interface to cut the finite elements. On the noninterface element, the standard Crouzeix–Raviart element and the P 0 {P_{0}} element pair is used. On the interface element, the basis functions developed for scalar interface problems (Kwak et al., An analysis of a broken P 1 {P_{1}} -nonconforming finite element method for interface problems, SIAM J. Numer. Anal. (2010)) are modified in such a way that the coupling between the velocity and pressure variable is different. There are two kinds of basis functions. The first kind of basis satisfies the Laplace–Young condition under the assumption of the continuity of the pressure variable. In the second kind, the velocity is of bubble type and is coupled with the discontinuous pressure, still satisfying the Laplace–Young condition. We remark that in the second kind the pressure variable has two degrees of freedom on each interface element. Therefore, our methods can handle the discontinuous pressure case. Numerical results including the case of the discontinuous pressure variable are provided. We see optimal convergence orders for all examples.

We present two kinds of lowest-order virtual element methods for planar linear elasticity problems. For the first one we use the nonconforming virtual element method with a stabilizing term. It can be interpreted as a modification of the nonconforming Crouzeix-Raviart finite element method as suggested in [22] to the virtual element method. For the second one we use the conforming virtual element for one component of the displacement vector and the nonconforming virtual element for the other. This approach can be seen as an extension of the idea of Kouhia and Stenberg suggested in [23] to the virtual element method. We show that our proposed methods satisfy the discrete Korn's inequality. We also prove that the methods are convergent uniformly for the nearly incompressible case and the convergence rates are optimal.

The next 6th generation (6G) mobile communication technologies should solve the out-of-band emission problem of cyclic prefix (CP) orthogonal frequency division multiplexing (OFDM) that the 4th generation technology did not solve in order to use frequency resources efficiently. In this study, we propose the spectral encapsulation (SE) that shapes the spectrum of OFDM signals. We also derive the orthogonalizing matrix and thoroughly analyze its characteristic. Since vector (v) SE-OFDM is a spectral shaping technique based on zero insertion without using any long filter, it can be applied to even short OFDM symbols. We build the precoder with the accurate orthogonal matrix and show that it has perfect blocking at the frequency bandwidth boundaries. Since the information transmission part of the proposed vSE-OFDM is the same as the CP-OFDM, it maintains the bit error rate (BER) performance of CP-OFDM as it is, and the complexity can be significantly improved because it can be implemented using IFFT.

We develop a numerical method for elliptic interface problems with implicit jumps. To handle the discontinuity, we enrich usual P1-conforming finite element space by adding extra degrees of freedom on one side of the interface. Next, we define a new bilinear form, which incorporates the implicit jump conditions. We show that the bilinear form is coercive and bounded if the penalty term is sufficiently large. We prove the optimal error estimates in both energy-like norm and L2-norm. We provide numerical experiments. We observe that our scheme converges with optimal rates, which coincides with our error analysis.

We develop a numerical scheme for a two-phase immiscible flow in heterogeneous porous media using a structured grid finite element method, which has been successfully used for the computation of various physical applications involving elliptic interface equations (Li et al., 2003, 2004; Chou et al., 2010; Kwak et al., 2010; Chang and Kwak, 2011). The proposed method is based on the implicit pressure-explicit saturation procedure. To solve the pressure equation, we use an IFEM based on the Rannacher–Turek Rannacher and Turek (1992) nonconforming space, which is a modification of the work in Kwak et al. (2010) where ‘broken’ P 1 nonconforming element of Crouzeix–Raviart (1973) [1] was developed. For the Darcy velocity, we apply the mixed finite volume method studied in Chou et al. (2003) and Kwak et al. (2010) on the basis of immersed finite element method (IFEM). In this way, the Darcy velocity of the flow can be computed cheaply (locally) after we solve the pressure equation. The computed Darcy velocity is used to solve the saturation equation explicitly. The whole procedure can be implemented on a structured grid which is independent of the underlying heterogeneous porous media. In fact, we apply a new version of multigrid algorithm to solve the pressure equation for which the CPU time grows like O ( N ) . Numerical tests for analytic problems show that our method is almost optimal in all variables. Problem with sources/sinks is also computed and it seems to work well.

This paper introduces a novel AI framework for nationwide wireless KPI monitoring and its diagnosis in LTE/5G mobile networks. The proposed method, which has not been provided so far for a commercial use, automatically detects meaningful degradations of cell-specific wireless KPI statistics and also provides an alarm to administrator with possible causes of the corresponding problem. Machine learning and deep learning technologies are properly applied to detect relative KPI degradations on the basis of those estimated globa/local PDFs and also to carry out root cause analysis for the problem diagnosis, respectively. We have confirmed the feasibility of the mentioned AI framework by completing a first PoC through interworking our AI server with commercial LTE/5G networks. Currently, about 20 major monitoring KPIs such as RRE, ENDC,RSSI, BLER, and so on are being considered in current 5G NSA mode.

In this paper, we propose a finite element method for the elasticity problems which have displacement discontinuity along the material interface using uniform grids. We modify the immersed finite element method introduced recently for the computation of interface problems having homogeneous jumps [20, 22]. Since the interface is allowed to cut through the element, we modify the standard Crouzeix-Raviart basis functions so that along the interface, the normal stress is continuous and the jump of the displacement vector is proportional to the normal stress. We construct the broken piecewise linear basis functions which are uniquely determined by these conditions. The unknowns are only associated with the edges of element, except the intersection points. Thus our scheme has fewer degrees of freedom than most of the XFEM type of methods in the existing literature [1,8,13]. Finally, we present numerical results which show optimal orders of convergence rates.

We develop a new finite element method for solving planar elasticity problems involving heterogeneous materials with a mesh not necessarily aligning with the interface of the materials. This method is based on the ‘broken’ Crouzeix–Raviart P 1 -nonconforming finite element method for elliptic interface problems [D.Y. Kwak, K.T. Wee and K.S. Chang, SIAM J. Numer. Anal. 48 (2010) 2117–2134]. To ensure the coercivity of the bilinear form arising from using the nonconforming finite elements, we add stabilizing terms as in the discontinuous Galerkin (DG) method [D.N. Arnold, SIAM J. Numer. Anal. 19 (1982) 742–760; D.N. Arnold and F. Brezzi, in Discontinuous Galerkin Methods. Theory, Computation and Applications , edited by B. Cockburn, G.E. Karniadakis, and C.-W. Shu. Vol. 11 of Lecture Notes in Comput. Sci. Engrg. Springer-Verlag, New York (2000) 89–101; M.F. Wheeler, SIAM J. Numer. Anal. 15 (1978) 152–161.]. The novelty of our method is that we use meshes independent of the interface, so that the interface may cut through the elements. Instead, we modify the basis functions so that they satisfy the Laplace–Young condition along the interface of each element. We prove optimal H 1 and divergence norm error estimates. Numerical experiments are carried out to demonstrate that our method is optimal for various Lame parameters μ and λ and locking free as λ → ∞.

In this paper, we propose a precoding scheme for 6th generation (6G) mobile communications, which can enhance the spectrum efficiency by overcoming the shortcoming of OFDM of power flowing out to adjacent channels and estimate the channel state with pilots. The proposed scheme is the first trial to have spectral efficiency by having orthogonality between subcarriers in the in-band as well as out-of-band (OOB) spectral efficiency. The precoding is realized with a matrix that consists of a cyclic prefix matrix, a central matrix, and a cyclic postfix matrix. The cyclic prefix and postfix matrices, and the central matrix play roles to provide the packet with the guard interval and to carry an information-bearing signal, respectively. The fade in and fade out properties of cyclic prefix and postfix can confine the spectrum of OFDM signals and prohibit the power leakage to adjacent channels. A good autocorrelation characteristic of the precoding scheme can guarantee the frequency offset estimation with high quality. Since the columns of the central matrix in the time domain are orthogonal to each other, the pilot symbols can be inserted in the same manner as in the conventional OFDM scheme.We will show that the proposed precoding scheme is useful in terms of out-of-band power leakage blocking, autocorrelation, and BER performance compared with OFDM.

An immersed nonconforming finite element method based on the flux continuity on intercell boundaries is introduced. The direct application of flux continuity across the support of basis functions yields a nonsymmetric stiffness system for interface elements. To overcome non-symmetry of the stiffness system we introduce a modification based on the Riesz representation and a local postprocessing to recover local fluxes. This approach yields a P 1 immersed nonconforming finite element method with a slightly different source term from the standard nonconforming finite element method. The recovered numerical flux conserves total flux in arbitrary sub-domain. An optimal rate of convergence in the energy norm is obtained and numerical examples are provided to confirm our analysis.

Two conditions are needed to solve numerical weather prediction models: initial condition and boundary condition. The initial condition has an especially important bearing on the model performance. To get a good initial condition, many data assimilation techniques have been developed for the meteorological and the oceanographical fields. Currently, the most commonly used technique for operational applications is the 3 dimensional (3-D) or 4 dimensional variational data assimilation method. The numerical method used for the cost function minimising process is usually an iterative method such as the conjugate gradient. In this paper, we use the multigrid method based on the cell-centred finite difference on the variational data assimilation to improve the performance of the minimisation procedure for 3D-Var data assimilation. Keywords : numerical weather prediction, variational data assimilation, minimization procedure, multigrid methods, cell centered finite difference (Published: 8 July 2014) Citation : Tellus A 2014, 66 , 20217, http://dx.doi.org/10.3402/tellusa.v66.20217

We introduce a new class of higher order mixed finite volume methods for elliptic problems. We start from the usual way of changing the given equation into a mixed system using the Darcy's law, ${\bf u}=-{\cal K} \nabla p$. By integrating the system of equations with some judiciously chosen test spaces on each element, we define new mixed finite volume methods of higher order. We show that these new schemes are equivalent to the nonconforming finite element spaces used to define them. The Darcy velocity can be locally recovered from the solution of nonconforming finite element method. Hence our work opens a way to make higher order mixed method more practicable. Three-dimensional extensions to parallelepiped elements are also presented. This work can be viewed as a generalization of earlier works which were devoted to reducing the (lowest order) mixed finite element method to a corresponding nonconforming finite element method. An optimal error analysis is carried out and numerical results are presented w...

We study a new mixed finite element of lowest order for general quadrilateral grids which gives optimal order error in the H (div)-norm. This new element is designed so that the H (div)-projection Π h satisfies ∇ · Π h = P h div. A rigorous optimal order error estimate is carried out by proving a modified version of the Bramble–Hilbert lemma for vector variables. We show that a local H (div)-projection reproducing certain polynomials suffices to yield an optimal L 2 -error estimate for the velocity and hence our approach also provides an improved error estimate for original Raviart–Thomas element of lowest order. Numerical experiments are presented to verify our theory.

We propose a new numerical method to solve an elliptic problem with jumps both in the solution and derivative along an interface. By considering a suitable function which has the same jumps as the solution, we transform the problem into one without jumps. Then we apply the immersed finite element method in which we allow uniform meshes so that the interface may cut through elements to discretize the problem as introduced in [1] , [2] , [3] . Some convenient way of approximating the jumps of the solution by piecewise linear functions is suggested. Our method can also handle the case when the interface passes through grid points. We believe this paper presents the first resolution of such cases. Numerical experiments for various problems show second-order convergence in L2 and first order in H1-norms. Moreover, the convergence order is very robust for all problems tested.

We discuss the general energetic variational approaches for hydrodynamic systems of complex fluids. In these energetic variational approaches, the least action principle (LAP) with action functional gives the Hamiltonian parts (conservative force) of the hydrodynamic systems, and the maximum/minimum dissipation principle (MDP), i.e., Onsager's principle, gives the dissipative parts (dissipative force) of the systems. When we combine the two systems derived from the two different principles, we obtain a whole coupled nonlinear system of equations satisfying the dissipative energy law. We will discuss the important roles of MDP in designing numerical method for computations of hydrodynamic systems in complex fluids. We will reformulate the dissipation in energy equation in terms of a rate in time by using an appropriate evolution equations, then the MDP is employed in the reformulated dissipation to obtain the dissipative force for the hydrodynamic systems. The systems are consistent with the Hamiltonian parts which are derived from LAP. This procedure allows the usage of lower order element (a continuous $C^0$ finite element) in numerical method to solve the system rather than high order elements, and at the same time preserves the dissipative energy law. We also verify this method through some numerical experiments in simulating the free interface motion in the mixture of two different fluids.

We study some numerical methods for solving a second order elliptic problem with interface. We introduce an immersed finite element method based on the “broken” $P_1$-nonconforming piecewise linear polynomials on interface triangular elements having edge averages as degrees of freedom. These linear polynomials are broken to match the homogeneous jump condition along the interface which is allowed to cut through the element. We prove optimal orders of convergence in the $H^1$- and $L^2$-norm. Next we propose a mixed finite volume method in the context introduced in [S. H. Chou, D. Y. Kwak, and K. Y. Kim, Math. Comp., 72 (2003), pp. 525-539] using the Raviart-Thomas mixed finite element and this “broken” $P_1$-nonconforming element. The advantage of this mixed finite volume method is that once we solve the symmetric positive definite pressure equation (without Lagrangian multiplier), the velocity can be computed locally by a simple formula. This procedure avoids solving the saddle point problem. Furthermore, we show optimal error estimates of velocity and pressure in our mixed finite volume method. Numerical results show optimal orders of error in the $L^2$-norm and broken $H^1$-norm for the pressure and in the $H(\mathrm{div})$-norm for the velocity.

Recently, new higher order finite volume methods (FVM) were introduced in [Z. Cai, J. Douglas, M. Park, Development and analysis of higher order finite volume methods over rectangles for elliptic equations, Adv. Comput. Math. 19 (2003) 3–33], where the linear system derived by the hybridization with Lagrange multiplier satisfying the flux consistency condition is reduced to a linear system for a pressure variable by an appropriate quadrature rule. We study the convergence of an iterative solver for this linear system. The conjugate gradient (CG) method is a natural choice to solve the system, but it seems slow, possibly due to the non-diagonal dominance of the system. In this paper, we propose block iterative methods with a reordering scheme to solve the linear system derived by the higher order FVM and prove their convergence. With a proper ordering, each block subproblem can be solved by fast methods such as the multigrid (MG) method. The numerical experiments show that these block iterative methods are much faster than CG.

We analyze an immersed interface finite element method based on linear polynomials on noninterface triangular elements and piecewise linear polynomials on interface triangular elements. The flux jump condition is weakly enforced on the smooth interface. Optimal error estimates are derived in the broken H 1-norm and L 2-norm.