A sharp interface method using enriched finite elements for elliptic interface problems

We present an immersed boundary method for the solution of elliptic interface problems with discontinuous coefficients which provides a second-order approximation of the solution. The proposed method can be categorised as an extended or enriched finite element method. In contrast to other extended FEM approaches, the new shape functions get projected in order to satisfy the Kronecker-delta property with respect to the interface. The resulting combination of projection and restriction was already derived in Höllbacher and Wittum (TBA, 2019a) for application to particulate flows. The crucial benefits are the preservation of the symmetry and positive definiteness of the continuous bilinear operator. Besides, no additional stabilisation terms are necessary. Furthermore, since our enrichment can be interpreted as adaptive mesh refinement, the standard integration schemes can be applied on the cut elements. Finally, small cut elements do not impair the condition of the scheme and we propose a simple procedure to ensure good conditioning independent of the location of the interface. The stability and convergence of the solution will be proven and the numerical tests demonstrate optimal order of convergence.

Central discontinuous Galerkin methods on overlapping meshes for wave equations

In this paper, we study the central discontinuous Galerkin (DG) method on overlapping meshes for second order wave equations. We consider the first order hyperbolic system, which is equivalent to the second order scalar equation, and construct the corresponding central DG scheme. We then provide the stability analysis and the optimal error estimates for the proposed central DG scheme for one- and multi-dimensional cases with piecewise ${P}^k$ elements. The optimal error estimates are valid for uniform Cartesian meshes and polynomials of arbitrary degree $k\geq 0$. In particular, we adopt the techniques in \cite{liu2018central, liu2020pk} and obtain the local projection that is crucial in deriving the optimal order of convergence. The construction of the projection here is more challenging since the unknowns are highly coupled in the proposed scheme. Dispersion analysis is performed on the proposed scheme for one dimensional problems, indicating that the numerical solution with $P^1$ elements reaches its minimum with a suitable parameter in the dissipation term. Several numerical examples including accuracy tests and long time simulation are presented to validate the theoretical results.

A Reduced Crouzeix–Raviart Immersed Finite Element Method for Elasticity Problems with Interfaces

The purpose of this paper is to develop a reduced Crouzeix–Raviart immersed finite element method (RCRIFEM) for two-dimensional elasticity problems with interface, which is based on the Kouhia–Stenberg finite element method (Kouhia et al. 1995) and Crouzeix–Raviart IFEM (CRIFEM) (Kwak et al. 2017). We use a {P_{1}}-conforming like element for one of the components of the displacement vector, and a {P_{1}}-nonconforming like element for the other component. The number of degrees of freedom of our scheme is reduced to two thirds of CRIFEM. Furthermore, we can choose penalty parameters independent of the Poisson ratio. One of the penalty parameters depends on Lamé’s second constant μ, and the other penalty parameter is independent of both μ and λ. We prove the optimal order error estimates in piecewise {H^{1}}-norm, which is independent of the Poisson ratio. Numerical experiments show optimal order of convergence both in {L^{2}} and piecewise {H^{1}}-norms for all problems including nearly incompressible cases.

On Dynamics of Iterative Techniques for Nonlinear Equation with Applications in Engineering

In this article, we construct an optimal family of iterative methods for finding the single root and then extend this family for determining all the distinct as well as multiple roots of single-variable nonlinear equations simultaneously. Convergence analysis is presented for both the cases to show that the optimal order of convergence is 4 in the case of single root finding methods and 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The computational cost, basins of attraction, efficiency, log of residual, and numerical test examples show that the newly constructed methods are more efficient as compared to the existing methods in the literature.

New Modification of Behl's Method Free from Second Derivative with an Optimal Order of Convergence

Behl’s method is one of the iterative methods to solve a nonlinear equation that converges cubically. In this paper, we modified the iterative method with real parameter β using second Taylor’s series expansion and reduce the second derivative of the proposed method using the equality of Chun-Kim and Newton Steffensen. The result showed that the proposed method has a fourth-order convergence for b = 0 and involves three evaluation functions per iteration with the efficiency index equal to 41/3 = 1.5874. Numerical simulation is presented for several functions to demonstrate the performance of the new method. The final results show that the proposed method has better performance as compared to some other iterative methods.Keywords: efficiency index; third-order iterative method; Chun-Kim’s method; Newton-Steffensen’s method; nonlinear equation. AbstrakMetode Behl adalah salah satu metode iterasi yang digunakan untuk menyelesaikan persamaan nonlinear dengan orde konvergensi tiga. Pada artikel ini, modifikasi terhadap metode iterasi menggunakan ekspansi deret Taylor orde dua dengan parameter β dan turunan kedua dihilangkan menggunakan penyetaraan dari metode Chun-Kim dan Newton-Steffensen. Hasil kajian menunjukkan bahwa metode iterasi yang diusulkan memiliki orde konvergensi empat untuk b = 0 dan melibatkan tiga evaluasi fungsi setiap iterasinya dengan indeks efisiensi sebesar 41/3 = 1,5874. Simulasi numerik dilakukan terhadap beberapa fungsi untuk menunjukkan performa modifikasi metode iterasi yang diusulkan. Hasil akhir menunjukkan bahwa metode iterasi tersebut mempunyai performa lebih baik dibandingkan dengan beberapa metode iterasi lainnya.Kata kunci: indeks efisiensi; metode iterasi orde tiga; metode Chun-Kim; metode Newton- Steffensen; persamaan nonlinear.

Cell-Centered Nonlinear Finite-Volume Methods With Improved Robustness

Summary We present a nonlinear finite-volume method (NFVM) that is either positivity-preserving or extremum-preserving with improved robustness. The key ingredient of the method is the construction of one-sided fluxes, which involves decomposition of conormal vectors by introducing harmonic-averaging points as auxiliary points. The original NFVM using harmonic-averaging points is not robust in the sense that decomposition of conormal vectors with nonnegative coefficients can easily run into difficulties for heterogeneous and anisotropic permeability tensors on general nonorthogonal meshes. To improve NFVM robustness, we first present an alternative derivation of harmonic-averaging points and give a different formula that shows more clearly a point's location. On the basis of the derivation of the new formula, a correction algorithm is proposed to make modifications to those problematic harmonic-averaging points so that all the conormal vectors can be decomposed with nonnegative coefficients successfully. As a result, the resulting NFVM can be applied to more-challenging problems when conormal decomposition with nonnegative coefficients is not possible without correction. The correction algorithm is a compromise between robustness and accuracy. While it improves the robustness of the resulting NFVM, results of numerical convergence tests show that the effect of our correction algorithm on accuracy is problem-dependent. Optimal order of convergence is still maintained for some problems, and the convergence rate is reduced for others. Monotonicity and extremum-preserving properties are verified by numerical experiments. Finally, a field test case is used to demonstrate that the NFVM combined with our correction algorithm can be applied to simulate real-life reservoirs of industry-standard complexity.

Optimal Derivative-Free Root Finding Methods Based on Inverse Interpolation

Finding a simple root for a nonlinear equation f ( x ) = 0 , f : I ⊆ R → R has always been of much interest due to its wide applications in many fields of science and engineering. Newton’s method is usually applied to solve this kind of problems. In this paper, for such problems, we present a family of optimal derivative-free root finding methods of arbitrary high order based on inverse interpolation and modify it by using a transformation of first order derivative. Convergence analysis of the modified methods confirms that the optimal order of convergence is preserved according to the Kung-Traub conjecture. To examine the effectiveness and significance of the newly developed methods numerically, several nonlinear equations including the van der Waals equation are tested.

Analysis of an adaptive HDG method for the Brinkman problem

We introduce and analyze a hybridizable discontinuous Galerkin method for the gradient-velocity-pressure formulation of the Brinkman problem. We present an a priori error analysis of the method, showing optimal order of convergence of the error. We also introduce an a posteriori error estimator, of the residual type, which helps us to improve the quality of the numerical solution. We establish reliability and local efficiency of our estimator for the $L^{2} $-error of the velocity gradient and the pressure and the $ H^{1} $-error of the velocity, with constants written explicitly in terms of the physical parameters and independent of the size of the mesh. In particular, our results are also valid for the Stokes problem. Finally, we provide numerical experiments showing the quality of our adaptive scheme.

Design and Analysis of a New Class of Derivative-Free Optimal Order Methods for Nonlinear Equations

We develop a general class of derivative free iterative methods with optimal order of convergence in the sense of Kung–Traub hypothesis for solving nonlinear equations. The methods possess very simple design, which makes them easy to remember and hence easy to implement. The Methodology is based on quadratically convergent Traub–Steffensen scheme and further developed by using Padé approximation. Local convergence analysis is provided to show that the iterations are locally well defined and convergent. Numerical examples are provided to confirm the theoretical results and to show the good performance of new methods.