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Adaptive Multi-level Algorithm for a Class of Nonlinear Problems Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-02-27 Dongho Kim, Eun-Jae Park, Boyoon Seo
In this article, we propose an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems. We have shown that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton’s method across different mesh levels, which is numerically validated. Our framework facilitates to use the general theory established
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A Phase-Space Discontinuous Galerkin Scheme for the Radiative Transfer Equation in Slab Geometry Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-02-20 Riccardo Bardin, Fleurianne Bertrand, Olena Palii, Matthias Schlottbom
We derive and analyze a symmetric interior penalty discontinuous Galerkin scheme for the approximation of the second-order form of the radiative transfer equation in slab geometry. Using appropriate trace lemmas, the analysis can be carried out as for more standard elliptic problems. Supporting examples show the accuracy and stability of the method also numerically, for different polynomial degrees
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Convergence of Adaptive Crouzeix–Raviart and Morley FEM for Distributed Optimal Control Problems Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-02-09 Asha K. Dond, Neela Nataraj, Subham Nayak
This article discusses the quasi-optimality of adaptive nonconforming finite element methods for distributed optimal control problems governed by 𝑚-harmonic operators for m = 1 , 2 m=1,2 . A variational discretization approach is employed and the state and adjoint variables are discretized using nonconforming finite elements. Error equivalence results at the continuous and discrete levels lead to
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Discontinuous Galerkin Methods for the Vlasov–Stokes System Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-01-30 Harsha Hutridurga, Krishan Kumar, Amiya K. Pani
This paper develops and analyses a semi-discrete numerical method for the two-dimensional Vlasov–Stokes system with periodic boundary condition. The method is based on the coupling of the semi-discrete discontinuous Galerkin method for the Vlasov equation with discontinuous Galerkin scheme for the stationary incompressible Stokes equation. The proposed method is both mass and momentum conservative
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Robust Multigrid Methods for Discontinuous Galerkin Discretizations of an Elliptic Optimal Control Problem Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-01-30 Sijing Liu
We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem, and we propose multigrid methods to solve the discretized system. We prove that the 𝑊-cycle algorithm is uniformly convergent in the energy norm and is robust with respect to a regularization parameter on convex domains. Numerical results are shown for both 𝑊-cycle and 𝑉-cycle algorithms.
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Weak Convergence of the Rosenbrock Semi-implicit Method for Semilinear Parabolic SPDEs Driven by Additive Noise Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-01-30 Jean Daniel Mukam, Antoine Tambue
This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability
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Robust PRESB Preconditioning of a 3-Dimensional Space-Time Finite Element Method for Parabolic Problems Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-01-24 Ladislav Foltyn, Dalibor Lukáš, Marco Zank
We present a recently developed preconditioning of square block matrices (PRESB) to be used within a parallel method to solve linear systems of equations arising from tensor-product discretizations of initial boundary-value problems for parabolic partial differential equations. We consider weak formulations in Bochner–Sobolev spaces and tensor-product finite element approximations for the heat and
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Simultaneous Reconstruction of Speed of Sound and Nonlinearity Parameter in a Paraxial Model of Vibro-Acoustography in Frequency Domain Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-01-24 Barbara Kaltenbacher, Teresa Rauscher
In this paper, we consider the inverse problem of vibro-acoustography, a technique for enhancing ultrasound imaging by making use of nonlinear effects. It amounts to determining two spatially variable coefficients in a system of PDEs describing propagation of two directed sound beams and the wave resulting from their nonlinear interaction. To justify the use of Newton’s method for solving this inverse
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Machine Learning Estimators: Implementation and Comparison in Python Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-01-24 Fabian Merle
We compare different machine learning estimators and present details about their implementation in Python. The computational studies are conducted for classification as well as regression problems. Moreover, as one of the founding problems of machine learning, we present the specific classification task of handwritten digit recognition. In this connection, we discuss the mathematical formulation and
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Efficient P1-FEM for Any Space Dimension in Matlab Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-01-24 Stefanie Beuter, Stefan A. Funken
This paper deals with the efficient implementation of the finite element method with continuous piecewise linear functions (P1-FEM) in R d \mathbb{R}^{d} ( d ∈ N d\in\mathbb{N} ). Although at present there does not seem to be a very high practical demand for finite element methods that use higher-dimensional simplicial partitions, there are some advantages in studying the efficient implementation of
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Quasi-Optimality of an AFEM for General Second Order Elliptic PDE Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-01-05 Arnab Pal, Thirupathi Gudi
In this article, convergence and quasi-optimal rate of convergence of an adaptive finite element method (in short, AFEM) is shown for a general second-order non-selfadjoint elliptic PDE with convection term b ∈ [ L ∞ ( Ω ) ] d {b\in[L^{\infty}(\Omega)]^{d}} and using minimal regularity of the dual problem, i.e., the solution of the dual problem has only H 1 {H^{1}} -regularity, which extends the
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Numerical Approximation of Gaussian Random Fields on Closed Surfaces Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-01-04 Andrea Bonito, Diane Guignard, Wenyu Lei
We consider the numerical approximation of Gaussian random fields on closed surfaces defined as the solution to a fractional stochastic partial differential equation (SPDE) with additive white noise. The SPDE involves two parameters controlling the smoothness and the correlation length of the Gaussian random field. The proposed numerical method relies on the Balakrishnan integral representation of
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Wave Propagation in High-Contrast Media: Periodic and Beyond Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-01-01 Élise Fressart, Barbara Verfürth
This work is concerned with the classical wave equation with a high-contrast coefficient in the spatial derivative operator. We first treat the periodic case, where we derive a new limit in the one-dimensional case. The behavior is illustrated numerically and contrasted to the higher-dimensional case. For general unstructured high-contrast coefficients, we present the Localized Orthogonal Decomposition
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An Optimal Method for High-Order Mixed Derivatives of Bivariate Functions Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-01-01 Evgeniya V. Semenova, Sergiy G. Solodky
The problem of optimal recovering high-order mixed derivatives of bivariate functions with finite smoothness is studied. Based on the truncation method, an algorithm for numerical differentiation is constructed, which is order-optimal both in the sense of accuracy and in terms of the amount of involved Galerkin information. Numerical examples are provided to illustrate the fact that our approach can
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Adaptive Image Compression via Optimal Mesh Refinement Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2024-01-01 Michael Feischl, Hubert Hackl
The JPEG algorithm is a defacto standard for image compression. We investigate whether adaptive mesh refinement can be used to optimize the compression ratio and propose a new adaptive image compression algorithm. We prove that it produces a quasi-optimal subdivision grid for a given error norm with high probability. This subdivision can be stored with very little overhead and thus leads to an efficient
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A Posteriori Error Estimation for the Optimal Control of Time-Periodic Eddy Current Problems Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-11-27 Monika Wolfmayr
This work presents the multiharmonic analysis and derivation of functional type a posteriori estimates of a distributed eddy current optimal control problem and its state equation in a time-periodic setting. The existence and uniqueness of the solution of a weak space-time variational formulation for the optimality system and the forward problem are proved by deriving inf-sup and sup-sup conditions
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Discontinuous Galerkin Two-Grid Method for the Transient Navier–Stokes Equations Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-11-24 Kallol Ray, Deepjyoti Goswami, Saumya Bajpai
In this paper, we apply a two-grid scheme to the DG formulation of the 2D transient Navier–Stokes model. The two-grid algorithm consists of the following steps: Step 1 involves solving the nonlinear system on a coarse mesh with mesh size 𝐻, and Step 2 involves linearizing the nonlinear system by using the coarse grid solution on a fine mesh of mesh size ℎ and solving the resulting system to produce
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A 𝐶1-𝑃7 Bell Finite Element on Triangle Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-11-22 Xuejun Xu, Shangyou Zhang
We construct a C 1 C^{1} - P 7 P_{7} Bell finite element by restricting its normal derivative from a P 6 P_{6} polynomial to a P 5 P_{5} polynomial, and its second normal derivative from a P 5 P_{5} polynomial to a P 4 P_{4} polynomial, on the three edges of every triangle. On one triangle, the finite element space contains the P 6 P_{6} polynomial space. We show the method converges at order 7 in
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Optimal Pressure Recovery Using an Ultra-Weak Finite Element Method for the Pressure Poisson Equation and a Least-Squares Approach for the Gradient Equation Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-11-20 Douglas R. Q. Pacheco, Olaf Steinbach
Reconstructing the pressure from given flow velocities is a task arising in various applications, and the standard approach uses the Navier–Stokes equations to derive a Poisson problem for the pressure p. That method, however, artificially increases the regularity requirements on both solution and data. In this context, we propose and analyze two alternative techniques to determine p ∈ L 2 ( Ω )
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Adaptive Absorbing Boundary Layer for the Nonlinear Schrödinger Equation Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-10-31 Hans Peter Stimming, Xin Wen, Norbert J. Mauser
We present an adaptive absorbing boundary layer technique for the nonlinear Schrödinger equation that is used in combination with the Time-splitting Fourier spectral method (TSSP) as the discretization for the NLS equations. We propose a new complex absorbing potential (CAP) function based on high order polynomials, with the major improvement that an explicit formula for the coefficients in the potential
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A Conforming Virtual Element Method for Parabolic Integro-Differential Equations Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-10-11 Sangita Yadav, Meghana Suthar, Sarvesh Kumar
This article develops and analyses a conforming virtual element scheme for the spatial discretization of parabolic integro-differential equations combined with backward Euler’s scheme for temporal discretization. With the help of Ritz–Voltera and L 2 L^{2} projection operators, optimal a priori error estimates are established. Moreover, several numerical experiments are presented to confirm the computational
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A Time Splitting Method for the Three-Dimensional Linear Pauli Equation Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-10-05 Timon S. Gutleb, Norbert J. Mauser, Michele Ruggeri, Hans Peter Stimming
We analyze a numerical method to solve the time-dependent linear Pauli equation in three space dimensions. The Pauli equation is a semi-relativistic generalization of the Schrödinger equation for 2-spinors which accounts both for magnetic fields and for spin, with the latter missing in preceding numerical work on the linear magnetic Schrödinger equation. We use a four term operator splitting in time
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Convergence of the Incremental Projection Method Using Conforming Approximations Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-10-04 Robert Eymard, David Maltese
We prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier–Stokes equations, without any regularity assumption on the weak solution. The velocity and the pressure are discretized in conforming spaces, whose compatibility is ensured by the existence of an interpolator for regular functions which preserves approximate divergence-free properties
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Three Low Order H-Curl-Curl Finite Elements on Triangular Meshes Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-10-03 Shangyou Zhang
We construct three H-curl-curl finite elements. The P 2 P_{2} and P 3 P_{3} vector finite element spaces are both enriched by one common P 4 P_{4} bubble and their local degrees of freedom are 13 and 21, respectively. As there does not exist any P 1 P_{1} H-curl-curl conforming finite element, the P 1 P_{1} H-curl-curl nonconforming finite element is constructed with three additional P 4 P_{4} bubbles
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Relaxation Quadratic Approximation Greedy Pursuit Method Based on Sparse Learning Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-10-03 Shihai Li, Changfeng Ma
A high-performance sparse model is very important for processing high-dimensional data. Therefore, based on the quadratic approximate greed pursuit (QAGP) method, we can make full use of the information of the quadratic lower bound of its approximate function to get the relaxation quadratic approximate greed pursuit (RQAGP) method. The calculation process of the RQAGP method is to construct two inexact
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Novel Raviart–Thomas Basis Functions on Anisotropic Finite Elements Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-09-18 Fleurianne Bertrand
Recently, H ( div ) \mathbf{H}(\mathrm{div}) -conforming finite element families were proven to be successful on anisotropic meshes, with the help of suitable interpolation error estimates. In order to ensure corresponding large-scale computation, this contribution provides novel Raviart–Thomas basis functions, robust regarding the anisotropy of a given triangulation. This new set of basis functions
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A Discontinuous Galerkin and Semismooth Newton Approach for the Numerical Solution of Bingham Flow with Variable Density Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-08-24 Sergio González-Andrade, Paul E. Méndez Silva
This paper is devoted to the study of Bingham flow with variable density. We propose a local bi-viscosity regularization of the stress tensor based on a Huber smoothing step. Next, our computational approach is based on a second-order, divergence-conforming discretization of the Huber regularized Bingham constitutive equations, coupled with a discontinuous Galerkin scheme for the mass density. We take
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Nonlinear PDE Models in Semi-relativistic Quantum Physics Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-08-16 Jakob Möller, Norbert J. Mauser
We present the self-consistent Pauli equation, a semi-relativistic model for charged spin- 1 / 2 1/2 particles with self-interaction with the electromagnetic field. The Pauli equation arises as the O ( 1 / c ) O(1/c) approximation of the relativistic Dirac equation. The fully relativistic self-consistent model is the Dirac–Maxwell equation where the description of spin and the magnetic field arises
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Symmetrized Two-Scale Finite Element Discretizations for Partial Differential Equations with Symmetric Solutions Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-08-07 Pengyu Hou, Fang Liu, Aihui Zhou
In this paper, some symmetrized two-scale finite element methods are proposed for a class of partial differential equations with symmetric solutions. With these methods, the finite element approximation on a fine tensor-product grid is reduced to the finite element approximations on a much coarser grid and a univariant fine grid. It is shown by both theory and numerics including electronic structure
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Space-Time Approximation of Local Strong Solutions to the 3D Stochastic Navier–Stokes Equations Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-08-07 Dominic Breit, Alan Dodgson
We consider the 3D stochastic Navier–Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates for the energy error with respect to convergence in probability, that is convergence of order (up to) 1 in space and of order (up to) 1/2 in time. The result holds up to the possible blow-up
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A Convergent Entropy-Dissipating BDF2 Finite-Volume Scheme for a Population Cross-Diffusion System Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-08-07 Ansgar Jüngel, Martin Vetter
A second-order backward differentiation formula (BDF2) finite-volume discretization for a nonlinear cross-diffusion system arising in population dynamics is studied. The numerical scheme preserves the Rao entropy structure and conserves the mass. The existence and uniqueness of discrete solutions and their large-time behavior as well as the convergence of the scheme are proved. The proofs are based
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A Convenient Inclusion of Inhomogeneous Boundary Conditions in Minimal Residual Methods Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-07-28 Rob Stevenson
Inhomogeneous essential boundary conditions can be appended to a well-posed PDE to lead to a combined variational formulation. The domain of the corresponding operator is a Sobolev space on the domain Ω on which the PDE is posed, whereas the codomain is a Cartesian product of spaces, among them fractional Sobolev spaces of functions on ∂ Ω \partial\Omega . In this paper, easily implementable minimal
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Computational Multiscale Methods for Nondivergence-Form Elliptic Partial Differential Equations Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-07-24 Philip Freese, Dietmar Gallistl, Daniel Peterseim, Timo Sprekeler
This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit
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Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-07-24 Pauline Achieng, Fredrik Berntsson, Vladimir Kozlov
We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to
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Multivariate Analysis-Suitable T-Splines of Arbitrary Degree Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-07-19 Robin Hiniborch, Philipp Morgenstern
This paper defines analysis-suitable T-splines for arbitrary degree (including even and mixed degrees) and arbitrary dimension. We generalize the concept of anchor elements known from the two-dimensional setting, extend existing concepts of analysis-suitability and show their sufficiency for linearly independent T-splines.
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Simultaneous Recovery of Two Time-Dependent Coefficients in a Multi-Term Time-Fractional Diffusion Equation Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-07-11 Wenjun Ma, Liangliang Sun
This paper deals with an inverse problem on simultaneously determining a time-dependent potential term and a time source function from two-point measured data in a multi-term time-fractional diffusion equation. First we study the existence, uniqueness and some regularities of the solution for the direct problem by using the fixed point theorem. Then a nice conditional stability estimate of inversion
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A Multilevel Extension of the GDSW Overlapping Schwarz Preconditioner in Two Dimensions Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-07-11 Alexander Heinlein, Oliver Rheinbach, Friederike Röver
Multilevel extensions of overlapping Schwarz domain decomposition preconditioners of Generalized Dryja–Smith–Widlund (GDSW) type are considered in this paper. The original GDSW preconditioner is a two-level overlapping Schwarz domain decomposition preconditioner, which can be constructed algebraically from the fully assembled stiffness matrix. The FROSch software, which belongs to the ShyLU package
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An Adaptive Two-Grid Solver for DPG Formulation of Compressible Navier–Stokes Equations in 3D Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-07-11 Waldemar Rachowicz, Witold Cecot, Adam Zdunek
We present an overlapping domain decomposition iterative solver for linear systems resulting from the discretization of compressible viscous flows with the Discontinuous Petrov–Galerkin (DPG) method in three dimensions. It is a two-grid solver utilizing the solution on the auxiliary coarse grid and the standard block-Jacobi iteration on patches of elements defined by supports of the coarse mesh base
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Approximate Deconvolution with Correction – A High Fidelity Model for Magnetohydrodynamic Flows at High Reynolds and Magnetic Reynolds Numbers Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-07-11 Yasasya Batugedara, Alexander E. Labovsky
We propose a model for magnetohydrodynamic flows at high Reynolds and magnetic Reynolds numbers. The system is written in the Elsässer variables so that the decoupling method of [C. Trenchea, Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows, Appl. Math. Lett. 27 (2014), 97–100] can be used. This decoupling method is only first-order accurate, so the proposed model
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Well-Posedness and Convergence Analysis of PML Method for Time-Dependent Acoustic Scattering Problems Over a Locally Rough Surface Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-07-07 Hongxia Guo, Guanghui Hu
We aim to analyze and calculate time-dependent acoustic wave scattering by a bounded obstacle and a locally perturbed non-selfintersecting curve. The scattering problem is equivalently reformulated as an initial-boundary value problem of the wave equation in a truncated bounded domain through a well-defined transparent boundary condition. Well-posedness and stability of the reduced problem are established
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A Domain Decomposition Scheme for Couplings between Local and Nonlocal Equations Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-06-19 Gabriel Acosta, Francisco M. Bersetche, Julio D. Rossi
We study a natural alternating method of Schwarz type (domain decomposition) for a certain class of couplings between local and nonlocal operators. We show that our method fits into Lions’s framework and prove, as a consequence, convergence in both the continuous and the discrete settings.
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Landweber Iterative Method for an Inverse Source Problem of Time-Space Fractional Diffusion-Wave Equation Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-06-14 Fan Yang, Yan Zhang, Xiao-Xiao Li
In this paper, we apply a Landweber iterative regularization method to determine a space-dependent source for a time-space fractional diffusion-wave equation from the final measurement. In general, this problem is ill-posed, and a Landweber iterative regularization method is used to obtain the regularization solution. Under the a priori parameter choice rule and the a posteriori parameter choice rule
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An Efficient Discretization Scheme for Solving Nonlinear Ill-Posed Problems Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-06-14 M. P. Rajan, Jaise Jose
Information based complexity analysis in computing the solution of various practical problems is of great importance in recent years. The amount of discrete information required to compute the solution plays an important role in the computational complexity of the problem. Although this approach has been applied successfully for linear problems, no effort has been made in literature to apply it to
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A Formulation for a Nonlinear Axisymmetric Magneto-Heat Coupling Problem with an Unknown Nonlocal Boundary Condition Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-06-14 Ran Wang, Huai Zhang, Tong Kang
This paper investigates a nonlinear axisymmetric magneto-heat coupling problem described by the quasi-static Maxwell’s equations and a heat equation. The coupling between them is provided through the temperature-dependent electric conductivity. The behavior of the material is defined by an anhysteretic 𝑯-𝑩 curve. The magnetic flux across a meridian section of the medium gives rise to the magnetic
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Identification of an Inverse Source Problem in a Fractional Partial Differential Equation Based on Sinc-Galerkin Method and TSVD Regularization Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-06-05 Ali Safaie, Amir Hossein Salehi Shayegan, Mohammad Shahriari
In this paper, using Sinc-Galerkin method and TSVD regularization, an approximation of the quasi-solution to an inverse source problem is obtained. To do so, the solution of direct problem is obtained by the Sinc-Galerkin method, and this solution is applied in a least squares cost functional. Then, to obtain an approximation of the quasi-solution, we minimize the cost functional by TSVD regularization
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A Second-Order Difference Scheme for Generalized Time-Fractional Diffusion Equation with Smooth Solutions Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-05-22 Aslanbek Khibiev, Anatoly Alikhanov, Chengming Huang
In the current work, we build a difference analog of the Caputo fractional derivative with generalized memory kernel (𝜇L2-1𝜎 formula). The fundamental features of this difference operator are studied, and on its ground, some difference schemes generating approximations of the second order in time for the generalized time-fractional diffusion equation with variable coefficients are worked out. We
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On a Mixed FEM and a FOSLS with 𝐻−1 Loads Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-05-08 Thomas Führer
We study variants of the mixed finite element method (mixed FEM) and the first-order system least-squares finite element (FOSLS) for the Poisson problem where we replace the load by a suitable regularization which permits to use H − 1 H^{-1} loads. We prove that any bounded H − 1 H^{-1} projector onto piecewise constants can be used to define the regularization and yields quasi-optimality of the lowest-order
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Volume Integral Equations and Single-Trace Formulations for Acoustic Wave Scattering in an Inhomogeneous Medium Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-05-05 Ignacio Labarca, Ralf Hiptmair
We study frequency domain acoustic scattering at a bounded, penetrable, and inhomogeneous obstacle Ω − ⊂ R d \Omega^{-}\subset\mathbb{R}^{d} , d = 2 , 3 d=2,3 . By defining constant reference coefficients, a representation formula for the pressure field is derived. It contains a volume integral operator, related to the one in the Lippmann–Schwinger equation. Besides, it features integral operators
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A Cost-Efficient Space-Time Adaptive Algorithm for Coupled Flow and Transport Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-05-02 Marius Paul Bruchhäuser, Markus Bause
In this work, a cost-efficient space-time adaptive algorithm based on the Dual Weighted Residual (DWR) method is developed and studied for a coupled model problem of flow and convection-dominated transport. Key ingredients are a multirate approach adapted to varying dynamics in time of the subproblems, weighted and non-weighted error indicators for the transport and flow problem, respectively, and
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A New Immersed Finite Element Method for Two-Phase Stokes Problems Having Discontinuous Pressure Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-04-26 Gwanghyun Jo, Do Young Kwak
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
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Implicit Runge–Kutta Schemes for Optimal Control Problems with Evolution Equations Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-03-31 Thomas G. Flaig
In this paper we discuss the use of implicit Runge–Kutta schemes for the time discretization of optimal control problems with evolution equations. The specialty of the considered discretizations is that the discretizations schemes for the state and adjoint state are chosen such that discretization and optimization commute. It is well known that for Runge–Kutta schemes with this property additional
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Implicit-Explicit Finite Difference Approximations of a Semilinear Heat Equation with Logarithmic Nonlinearity Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-03-30 Panagiotis Paraschis, Georgios E. Zouraris
We formulate an initial and Dirichlet boundary value problem for a semilinear heat equation with logarithmic nonlinearity over a two-dimensional rectangular domain. We approximate its solution by employing the standard second-order finite difference method for space discretization, and a linearized backward Euler method, or, a linearized BDF2 method for time stepping. For the linearized backward Euler
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Guaranteed Lower Eigenvalue Bounds for Steklov Operators Using Conforming Finite Element Methods Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-03-28 Taiga Nakano, Qin Li, Meiling Yue, Xuefeng Liu
For the eigenvalue problem of the Steklov differential operator, an algorithm based on the conforming finite element method (FEM) is proposed to provide guaranteed lower bounds for the eigenvalues. The proposed lower eigenvalue bounds utilize the a priori error estimation for FEM solutions to non-homogeneous Neumann boundary value problems, which is obtained by constructing the hypercircle for the
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Developments on the Stability of the Non-symmetric Coupling of Finite and Boundary Elements Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-03-09 Matteo Ferrari
We consider the non-symmetric coupling of finite and boundary elements to solve second-order nonlinear partial differential equations defined in unbounded domains. We present a novel condition that ensures that the associated semi-linear form induces a strongly monotone operator, keeping track of the dependence on the linear combination of the interior domain equation with the boundary integral one
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Fast Barrier Option Pricing by the COS BEM Method in Heston Model (with Matlab Code) Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-03-09 Alessandra Aimi, Chiara Guardasoni, Luis Ortiz-Gracia, Simona Sanfelici
In this work, the Fourier-cosine series (COS) method has been combined with the Boundary Element Method (BEM) for a fast evaluation of barrier option prices. After a description of its use in the Black and Scholes (BS) model, the focus of the paper is on the application of the proposed methodology to the barrier option evaluation in the Heston model, where its contribution is fundamental to improve
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CVEM-BEM Coupling for the Simulation of Time-Domain Wave Fields Scattered by Obstacles with Complex Geometries Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-03-08 Luca Desiderio, Silvia Falletta, Matteo Ferrari, Letizia Scuderi
In this paper, we present a numerical method based on the coupling between a Curved Virtual Element Method (CVEM) and a Boundary Element Method (BEM) for the simulation of wave fields scattered by obstacles immersed in homogeneous infinite media. In particular, we consider the 2D time-domain damped wave equation, endowed with a Dirichlet condition on the boundary (sound-soft scattering). To reduce
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Coupling of Finite and Boundary Elements for Singularly Nonlinear Transmission and Contact Problems Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-03-08 Heiko Gimperlein, Ernst P. Stephan
This article discusses the well-posedness and error analysis of the coupling of finite and boundary elements for interface problems in nonlinear elasticity. It concerns 𝑝-Laplacian-type Hencky materials with an unbounded stress-strain relation, as they arise in the modelling of ice sheets, non-Newtonian fluids or porous media. We propose a functional analytic framework for the numerical analysis and
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Force Computation for Dielectrics Using Shape Calculus Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-03-07 Piyush Panchal, Ning Ren, Ralf Hiptmair
We are concerned with the numerical computation of electrostatic forces/torques in only piece-wise homogeneous materials using the boundary element method (BEM). Conventional force formulas based on the Maxwell stress tensor yield functionals that fail to be continuous on natural trace spaces. Thus their use in conjunction with BEM incurs slow convergence and low accuracy. We employ the remedy discovered
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On Split Monotone Variational Inclusion Problem with Multiple Output Sets with Fixed Point Constraints Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-02-28 Victor Amarachi Uzor, Timilehin Opeyemi Alakoya, Oluwatosin Temitope Mewomo
In this paper, we introduce and study the concept of split monotone variational inclusion problem with multiple output sets (SMVIPMOS). We propose a new iterative scheme, which employs the viscosity approximation technique for approximating the solution of the SMVIPMOS with fixed point constraints of a nonexpansive mapping in real Hilbert spaces. The proposed method utilises the inertial technique
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The Tracking of Derivative Discontinuities for Delay Fractional Equations Based on Fitted L1 Method Comput. Methods Appl. Math. (IF 1.3) Pub Date : 2023-02-27 Dakang Cen, Seakweng Vong
In this paper, the analytic solution of the delay fractional model is derived by the method of steps. The theoretical result implies that the regularity of the solution at s + {s^{+}} is better than that at 0 + {0^{+}} , where s is a constant time delay. The behavior of derivative discontinuity is also discussed. Then, improved regularity solution is obtained by the decomposition technique and a fitted