The classical dual mixed finite element method for flow simulations is based on H(div,Ω) conforming approximation spaces for the flux, which guarantees continuous normal components on element interfaces, and discontinuous approximations in L2(Ω) for the pressure. However, stability and convergence can only be obtained for compatible approximation spaces. Stabilized finite element methods may provide

In this paper, an efficient linear compact alternating direction implicit (CADI) scheme with second order temporal accuracy and fourth order spatial accuracy is proposed for approximating solution to the initial–boundary problem for the two-dimensional Klein–Gordon–Schrödinger equations. Furthermore, maximum norm error estimates of numerical solutions are obtained by using the energy method and the

Under-resolved direct numerical simulations of turbulence in the wake of a single-square-grid (SSG) are conducted using the lattice-Boltzmann method (LBM), with varying degrees of partial moment entropic stabilisation. Turbulence statistics along the grid centreline show satisfactory agreement with literature, greatly outperforming a control study employing the single-relaxation-time Smagorinsky LES

In this study, we present a new type of domain decomposition algorithm to obtain the ground state solution for Bose–Einstein condensates. Using our proposed algorithm, instead of directly solving the nonlinear eigenvalue problem, one only needs to solve a series of linear boundary value problems on a finite element space sequence using the domain decomposition method, and subsequently solve a small-scale

The ADI type finite volume scheme is constructed to solve the non-classical heat conduction equation. The original 3D model in a tube is reduced to a hybrid dimension model in a large part of the domain. Special interface conditions are defined between 3D and 1D parts. It is assumed that the solution satisfies radial symmetry conditions in 3D parts. The finite volume method is applied to approximate

We propose and analyze a low-order virtual element for linear elasticity problems. It can be seen as an evolution of the Bernardi–Raugel element to general polygonal meshes. Like the Bernardi–Raugel element, the local degrees of freedoms for the new virtual element are the values of two components of v at vertexes and the values of the lowest moment of normal component of v on each edge. The proposed

In this paper, a numerical scheme is prepared using the higher-order shear deformation type of kinematic model with the help of a coupled finite and boundary elements (FE–BE) to evaluate the vibroacoustic responses of laminated composite sandwich curved shell panels. The panel is under the influence of a harmonic point load. Further, an FE–BE combined technique is utilized to prepare a generic computer

In this paper we present an a posteriori error analysis of a mixed-VEM discretization for a nonlinear Brinkman model of porous media flow, which has been proposed by the authors in a previous work. Therein, the system is formulated in terms of a pseudostress tensor and the velocity gradient, whereas the velocity and the pressure of the fluid are computed via postprocessing formulas. Furthermore, the

Space and time approximations for two-dimensional space fractional complex Ginzburg–Landau equation are examined. The schemes under consideration are discreted by the second-order backward differential formula (BDF2) in time and two classes of the fractional centered finite difference methods in space. A linearized technique is employed by the extrapolation. We prove the unique solvability and stability

For a wide range of PDEs, the discontinuous Petrov–Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan provides discrete stability starting from a coarse mesh and minimization of the residual in a user-controlled norm, among other appealing features. Research on DPG for transient problems has mainly focused on spacetime discretizations, which has theoretical advantages, but practical costs for

In the present study, we explore mechanical response of a radially constrained elastic porous shell during the passage of charged fluid (electrically conducting fluid). A constant magnetic field is exposed on the binary mixture of fluid and solid. The governing dynamics involved in the motion of fluid as well as solid deformation was based on the rate of applied compression at the inner radius of the

We derive formulas extracting stress intensity factors of the biharmonic equations on cracked domains with clamped (or simply supported or free) boundary conditions along the crack faces. Each of these formulas can be written in terms of the integral of the given source function multiplied by the cut-off dual singularity and the integral of the unknown true solution multiplied by the cut-off dual singularity

In this article, an iterative method suitable for inverting semilinear problems is presented. The method employs a two-way parametrization that is able to reduce semilinear boundary-value problems into simpler and linear systems governed by the Laplacian. In contrast to Newton–Picard iterations, the method prescribes a well-defined procedure for constructing initial iterates, and thus, does not suffer

This paper introduces a novel method for deducing high-order finite difference schemes for the three-dimensional Helmholtz equation, Δp±λ2p=f. The new method can reach an eighth-order of accuracy. It is based on an implicit formulation by computing simultaneously the unknown variable and its corresponding derivatives. The scheme is obtained from Taylor series expansion and plane wave theory analysis

We present a displacement-based and a mixed isogeometric collocation (IGA-C) formulation for free-form, three-dimensional, shear-deformable beams with high and rapidly-varying curvature and torsion. When such complex shapes are concerned, the approach used to build the IGA geometric model becomes relevant. Although IGA-C has been so far successfully applied to a wide range of problems, the effects

In this work, we introduce a novel abstract framework for the stability and convergence analysis of fully coupled discretisations of the poroelasticity problem and apply it to the analysis of Hybrid High-Order (HHO) schemes. A relevant feature of the proposed framework is that it rests on mild time regularity assumptions that can be derived from an appropriate weak formulation of the continuous problem

We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We perform a posteriori error analysis of Galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations. Practical

In this paper, a new technique is shown for deriving computable, guaranteed lower bounds of functional type (minorants) for two different cost functionals subject to a parabolic time-periodic boundary value problem. Together with previous results on upper bounds (majorants) for one of the cost functionals, both minorants and majorants lead to two-sided estimates of functional type for the optimal control

Relaxation models for the pressureless gas dynamics (PGD) equations attempt to satisfy the strictly hyperbolic conservation law in order to employ the well-posed approximated Riemann solvers. In this study, a new type of relaxation model is proposed to resolve two shortcomings of the existing relaxation models: the constant propagation speed of sound, and the collapse of delta shock waves in multidimensional

This report presents defect correcting extrapolation technique (DCE), an alternative approach for defect correction (DC). In contrast to conventional DC methods, which always solve for separate correction steps, DCE instantly assumes weighted linear combinations of several artificial viscosity approximations as a corrected solution. This way, it merely requires solving an identical problem with varying

The Open Porous Media (OPM) initiative is a community effort that encourages open innovation and reproducible research for simulation of porous media processes. OPM coordinates collaborative software development, maintains and distributes open-source software and open data sets, and seeks to ensure that these are available under a free license in a long-term perspective. In this paper, we present OPM

The paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi-static linear Biot equations. The system is decoupled by employing the fixed-stress split scheme, which leads to an iteratively solved semi-discrete system. The error bounds are derived by combining a posteriori estimates for contractive mappings

The modeling of turbulent flows is relevant in many engineering applications and, is an active field of research on numerical methods. Convergence and stability of proposed formulations are crucial to predict transitional flows from laminar to turbulent flows. In this work, a recently developed stabilized finite element formulation is used as a powerful tool to describe such kind of problems. The essential

This paper describes the development of the Method of Regularized Sources for potential flow problems. It is based on the modification of the fundamental solution near the source point by replacing the singularity with a blob in form of a steep rational function. This allows to solve the problems in the same way as with Method of Fundamental Solutions, however without an artificial boundary. Method

Thermally developing mixed electroosmotically and pressure-driven flow in a triangular micro-duct with a step change wall temperature is investigated. Both axial conduction (Graetz problem extended) and Joule heating effects are taken into consideration. A new higher order accurate method is proposed. The proposed method is based on pseudo spectral Galerkin approach and the eigenfunctions of the Helmholtz

In this work we present a Reduced Basis VMS-Smagorinsky Boussinesq model, applied to natural convection problems in a variable height cavity, in which the buoyancy forces are involved. We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, so as the height of the cavity. We perform an Empirical Interpolation Method to approximate

We present a FEM–BEM coupling strategy for time-harmonic acoustic scattering in media with variable sound speed. The coupling is realized with the aid of a mortar variable that is an impedance trace on the coupling boundary. The resulting sesquilinear form is shown to satisfy a Gårding inequality. Quasi-optimal convergence is shown for sufficiently fine meshes. Numerical examples confirm the theoretical

In this article, the condition number of the stiffness matrix κ(K) is compared for three high order finite element methods (FEMs), i.e., the p-version of the FEM, the spectral element method (SEM), and the NURBS-based isogeometric analysis (IGA). Note that only problems in linear elasticity are considered in the analysis. It is well-known that the condition number is one factor strongly influencing

Mesh-based numerical methods such as the Finite Element Method (FEM) and Finite Difference Method (FDM) are widely used in various fields of science and engineering. However, the shortcomings of mesh-based methods are becoming increasingly obvious mainly due to the distorted or broken elements when analyzing the problems of large deformation or crack propagation. To overcome the above problems, meshfree

In this paper, we put forward an efficient spectral-Galerkin approximation in view of dimension reduction scheme for transmission eigenvalue problem in polar geometries. Firstly, we turn the original problem into an equivalent fourth order nonlinear eigenvalue problem. Then the fourth order nonlinear eigenvalue problem is transformed into a coupled fourth order linear eigenvalue system by introducing

By introducing a regularization matrix and an additional iteration parameter, a new class of regularized deteriorated positive-definite and skew-Hermitian splitting (RDPSS) preconditioners are proposed for generalized saddle point linear systems. Compared with the well-known Hermitian and skew-Hermitian splitting (HSS) preconditioner and the regularized HSS preconditioner (Bai, 2019) studied recently

We prove a discrete comparison principle (equivalent to a discrete maximum principle) for the L1 discretisation of the Caputo time derivative and the standard 3-point discretisation of the spatial derivative in the time-fractional initial–boundary value problem Dtαu−pΔu+c(x,t)u=f, where p>0 but no assumption is made on the sign of c. (Previously, any discrete comparison principle relied on the assumption

In this paper, the unstructured mesh Galerkin finite element method with a weighted and shifted Grünwald difference approximation and Composite Trapezoid formula is presented to solve the nonhomogeneous two-dimensional distributed order time fractional Cable equation on irregular convex domains. The Crank–Nicolson type discretization of the finite element scheme is implemented to obtain the numerical

Utility-indifference approach is a useful approach to be adopted for pricing financial derivatives in an incomplete market and is an ongoing hot research topic in quantitative finance. One interesting question associated with this approach is whether or not it renders to the same option prices, degenerately, when the market becomes infinitesimally close to a complete market. The answer for such a question

A new weak Galerkin finite element method is introduced and analyzed for the Reissner–Mindlin plate model in the primary form (without introducing shear strain as an extra unknown), which results in a linear system with symmetric positive definite stiffness matrix. The proposed method achieves uniform convergence with respect to plate thickness (the so-called locking-free) without introducing any projection

The deflagration-to-detonation transition via the interaction of a weak shock with a series of discrete laminar flames is analyzed computationally based on the unsteady reactive Navier–Stokes equations with one-step Arrhenius chemistry. For comparison, simulations with the Euler equations are also performed. The numerical setup aims to mimic an array of laminar flames ignited at different spark times

We consider a chemo-repulsion model with quadratic production in a bounded domain. Firstly, we obtain global in time weak solutions, and give a regularity criterion (which is satisfied for 1D and 2D domains) to deduce uniqueness and global regularity. After, we study two cell-conservative and unconditionally energy-stable first-order time schemes: a (nonlinear and positive) Backward Euler scheme and

This work presents two new alternating direction implicit (ADI) schemes for solving parabolic interface problems in two and three dimensions. First, the ghost fluid method (GFM) is adopted for the first time in the literature to treat interface jumps in the ADI framework, which results in symmetric and tridiagonal finite difference matrices in each ADI step. The proposed GFM-ADI scheme achieves a first

This paper presents the mathematical formulation, numerical solution, calibration and testing of a physics-based model of wildfire propagation aimed at faster-than-real-time simulations. Despite a number of simplifying assumptions, the model is comprehensive enough to capture the major phenomena that govern the behaviour of a real fire — namely the pyrolysation of wood; the combustion of a mono-phase

This paper provides a rotational velocity-correction projection method for the unsteady incompressible magnetohydrodynamics (MHD) equations. In this method, the electrical field variable E was eliminated. In this way, the Gauss’s law can be conserved. The velocity variable u and the pressure field p are solved by a rotational velocity-correction projection method. The optimal error estimates of the

We consider an inverse problem where the forward problem is that of linear plane stress elasticity, or equivalently, that of linear heat/hydraulic conduction. We demonstrate that the linearized version of the saddle point problem obtained from the minimization problem inherits some stability from the forward elliptic problem. In particular, it is stable for the response variable and the Lagrange multiplier

In this paper, a non-uniform time-stepping convex-splitting numerical algorithm for solving the widely used time-fractional Cahn–Hilliard equation is introduced. The proposed numerical scheme employs the L1+ formula for discretizing the time-fractional derivative and a second-order convex-splitting technique to deal with the non-linear term semi-implicitly. Then the pseudospectral method is utilized

In this article we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convection-dominated case can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The idea of this article is to consider the approximation

In this paper, we make use of the fact that the matrix u is (approximately) low-rank in image inpainting, and the corresponding gradient transform matrices Dxu,Dyu are sparse in image reconstruction and restoration. Therefore we consider that these gradient matrices Dxu,Dyu also are (approximately) low-rank, and also verify it by numerical test and theoretical analysis. We propose a model called compressive

The software package BoSSS serves the discretization of (steady-state or time-dependent) partial differential equations with discontinuous coefficients and/or time-dependent domains by means of an eXtended Discontinuous Galerkin (XDG, resp. DG) method, aka. cut-cell DG or unfitted DG. This work consists of two major parts: First, the XDG method is introduced and a formal notation is developed, which

High order (HO) schemes are attractive candidates for the numerical solution of multiscale problems occurring in fluid dynamics and related disciplines. Among the HO discretization variants, discontinuous Galerkin schemes offer a collection of advantageous features which have lead to a strong increase in interest in them and related formulations in the last decade. The methods have matured sufficiently

OpenCAL is a scientific software library specifically developed for the simulation of 2D and 3D complex dynamical systems on parallel computational devices. It is written in C/C++ and relies on OpenMP/OpenCL and MPI for parallel execution on multi-/many-core devices and clusters of computers, respectively. The library provides the Extended Cellular Automata paradigm as a high-level formalism for modeling

We consider two-point, reaction–diffusion type, singularly perturbed boundary value problems of order 2ν∈Z+, and the approximation of their solution using isogeometric analysis. In particular, we use a Galerkin formulation with B-splines as basis functions, defined using appropriately chosen knot vectors. We prove robust exponential convergence in the energy norm, independently of the singular perturbation

To keep structures in the restoration problem is very important via coupling the local information of the image with the proposed model. In this paper we propose a local self-adaptive ℓp-regularization model for p∈(0,2) based on the total variation scheme, where the choice of p depends on the local structures described by the eigenvalues of the structure tensor. Since the proposed model as the classic

In this paper, we consider implicit and linearized dissipation-preserving Galerkin–Legendre spectral methods to solve space fractional nonlinear damped wave equation in two dimensions. The full discrete schemes preserve dissipation of energy in damped case and conservation of energy in undamped case. Moreover, the stability and convergence analysis of the full discrete schemes are rigorously given

The present work compares fluid–solid interface approaches for the lattice Boltzmann method (LBM) to study vortex-induced vibrations (VIV). Two classes of fluid–solid interface approaches, namely the partially saturated methods (PSM) (Noble and Torczynski (1998), Holdych (2003), Krause et al. (2017), Trunk et al. (2018)) and moving boundary methods (MBM) (Bouzidi et al. (2001), Lallemand and Luo (2003)

In this paper, we analyze the effects of vaccination from a spatial perspective. We propose a spatial deterministic SIRS–V model, which considers a non-linear system of partial differential equations with explicit attrition and diffusion terms for the vaccination process. The model allows us to simulate numerically the spatial and temporal dynamics of an epidemic, considering different spatial strategies

The peridynamic theory is advantageous for problems involving damage since the peridynamic equation of motion is valid everywhere, regardless of existing discontinuities, and an external criterion is not necessary for predicting damage initiation and propagation. However, the current peridynamic modeling methods utilize regular or quasi-regular lattice structures, which poses difficulties in modeling

We study a mathematical model of cancer cell invasion of tissue (extracellular matrix) consisting of a system of reaction–diffusion-taxis partial differential equations which describes the interactions between cancer cells, the matrix degrading enzyme and the host tissue. We analyze the local stability of the constant equilibrium solutions to the chemotaxis–haptotaxis system, we derive a discretization

In Lai and Friedman (2017) partial differential equations are used to model an inhibition of tumor applying the therapy of cancer combined with cancer vaccine plus checkpoint immune inhibitors. It is assumed there that the tumor is a spherical model and the treatment by the vaccine and immune checkpoint is not mathematically optimized. In this article we do not assume the tumor has a spherical shape

We present an implementation of a subgrid finite-volume model based on a nonlinear semi-implicit solver originally proposed by Casulli (2009), but with different approaches for constructing the tangential velocity as required by all Arakawa C-grid models. The model is carefully benchmarked through a few revealing test cases that consist of a theoretical nonlinear inundation case and a field application

This paper introduces LEoPart, an add-on for the open-source finite element software library FEniCS to seamlessly integrate Lagrangian particle functionality with (Eulerian) mesh-based finite element (FE) approaches. LEoPart- which is so much as to say: ‘Lagrangian–Eulerian on Particles’ - contains tools for efficient, accurate and scalable advection of Lagrangian particles on simplicial meshes. In

We present the OpenLB package, a C++ library providing a flexible framework for lattice Boltzmann simulations. The code is publicly available and published under GNU GPLv2, which allows for adaption and implementation of additional models. The extensibility benefits from a modular code structure achieved e.g. by utilizing template meta-programming. The package covers various methodical approaches and

For the iterative solution of finite element discretized, nonsmooth minimization problems the alternating direction method of multipliers (ADMM) is considered, which is a flexible method to solve a large class of convex minimization problems. Particular features are its unconditional convergence with respect to the involved step size and its direct applicability. In particular, this article deals with

This article describes a fast meshless technique, the direct meshless local Petrov–Galerkin (DMLPG) method, for identifying a control function in two-dimensional parabolic inverse PDEs. The DMLPG method is a truly meshless technique that directly approximates the local weak forms. In DMLPG, the local integrations are done over the low-degree polynomial basis functions instead of the complicated shape