In this paper, a second-order local and parallel space-time algorithm is proposed and analyzed for the heat equation. This scheme is based on the parareal with spectral deferred correction method in time and the expandable local and parallel finite element method in space. It realizes the parallelism both in the temporal as well as in the spatial direction. We prove its stability and the optimal error

We present an efficient second- or fourth-order finite difference direct numerical simulation (DNS) solver using pencil-like domain decomposition parallel strategy, with the ability to handle different boundary conditions. The viscous term is treated implicitly, partial implicitly, or explicitly. The runtimes for different viscous treatments and different boundary conditions are evaluated quantitatively

We propose an interpolation-based lattice Boltzmann formulation that is applicable to non-conforming orthogonal meshes. This interpolation configuration allows the use of localized and directional refinement that reduces mesh sizes and makes them well adapted for solving flows in various configurations, including aerodynamic shapes in free streams. The novel aspect of the proposed method lies in the

Full waveform inversion (FWI) entails the ill-posed reconstruction of material parameters (such as sound speed and attenuation) from measurements of complete wave fields (full seismograms). In this paper we present a novel framework for FWI in the visco-acoustic regime. The new framework is based on a new elegant derivation of the system of state and adjoint PDEs which are approximated by the discontinuous

A meshfree method on fixed grids is devised for simulation of Poisson's equation on 3D image-based cellular materials. The non-boundary fitted discretization of such jagged voxel models of complex geometries is accomplished through embedding the micro-CT scan image in a Cartesian grid of nodes. The computational nodes inside the solid voxels are found by a simple point-in-membership test. Using a set

In this paper, we consider an inverse problem for time-fractional advection-dispersion equation in a multi-layer composite medium. The main goal of our paper is to approximate the initial information, which is inaccessible for measurement, from the observation data at a certain point in second layer by constructing a regularized solution using a filter regularization method. Under appropriate regularity

Isogeometric Analysis (IGA) is a computational technique for the numerical approximation of partial differential equations (PDEs). This technique is based on the use of spline-type basis functions, that are able to hold a global smoothness and allow to exactly capture a wide set of common geometries. The current rise of this approach has encouraged the search of fast solvers for isogeometric discretizations

In this paper we study a class of quasi-variational-hemivariational inequalities in reflexive Banach spaces. The inequalities contain a convex potential, a locally Lipschitz superpotential, and a implicit obstacle set of constraints. Solution existence and compactness of the solution set to the inequality problem are established based on the Kakutani–Ky Fan fixed point theorem. The applicability of

This paper studies the numerical solution of a two-dimensional quenching type nonlinear reaction-diffusion problem via dimensional splitting. The variable coefficient differential equation considered possesses a nonlinear forcing term, and may lead to strong quenching singularities that have profound multiphysics and engineering applications to the energy industry. Our investigations focus on the construction

In this work, we consider the numerical computation for the two-dimensional generalized nonlinear Schrödinger equations with wave operator. Based on the scalar auxiliary variable (SAV) approach, the original problem is transformed into an equivalent one, which corresponds to the energy-conservation laws. We present an energy-conserving and linearly implicit three-level scheme for the equivalent system

In this article, we develop fourth-order compact difference schemes based on the linearized generalized BDF2-θ to solve the two-dimensional nonlinear fractional mobile/immobile (M/I) transport equations. We derive theoretical results, including unconditional stability and error estimates associated with solution regularity. Finally, we provide extensive numerical examples with smooth solutions to demonstrate

In simulation of heat conduction with temperature-independent physical properties and boundary conditions (BCs), Galerkin residual analysis and variational analysis yield equivalent finite element method (FEM), the conventional FEM. However, if the properties and BCs are temperature-dependent, it is discovered that their derivatives further induce nonlinearity of FEM which consequently generates divergence

The paper deals with the characterization of Powell-Sabin triangulations allowing the construction of bivariate quartic splines of class C2. The result is established by relating the triangle and edge split points provided by the refinement of each triangle. For a triangulation fulfilling the characterization obtained, a normalized representation of the splines in the C2 space is given.

We present a numerical approximation of Darcy's flow through a porous medium that incorporates networks of fractures with non empty intersection. Our scheme employs PolyDG methods, i.e. discontinuous Galerkin methods on general polygonal and polyhedral (polytopic, for short) grids, featuring elements with edges/faces that may be in arbitrary number (potentially unlimited) and whose measure may be arbitrarily

The advantages of using high-order time integration schemes for thermally coupled flows are assessed numerically. First-, second-, and third-order backward difference schemes are evaluated. The problem is solved in a decoupled manner using a nested iterative algorithm for the Navier–Stokes and energy equations to eliminate decoupling errors. For the space discretization, a stabilized finite element

We study linear quadratic Gaussian (LQG) control design for linear port-Hamiltonian systems. To this end, we exploit the freedom in choosing the weighting matrices and propose a specific choice which leads to an LQG controller which is port-Hamiltonian and, thus, in particular stable and passive. Furthermore, we construct a reduced-order controller via balancing and subsequent truncation. This approach

We present a low order virtual element discretization for time dependent Maxwell's equations, which allow for the use of general polyhedral meshes. Both the semi- and fully-discrete schemes are considered. We derive optimal a priori estimates and validate them on a set of numerical experiments. As pivot results, we discuss some novel inequalities associated with de Rahm sequences of nodal, edge, and

In this paper we use a modified constitutive law in a hereditary integral form to analyze the response of an isotropic non-uniform linear viscoelastic Timoshenko beam. A mixed method framework is used to provide stability and semi-discrete error estimates that do not deteriorate as the thickness parameter becomes small. Numerical experiments for both quasi-static and transient cases are presented.

For a tetrahedron, suppose that all internal angles of faces and all dihedral angles are less than a fixed constant C that is smaller than π. Then, it is said to satisfy the maximum angle condition with the constant C. The maximum angle condition is important in the error analysis of Lagrange interpolation on tetrahedrons. This condition ensures that we can obtain an error estimation, even on certain

Given two triangles in a planar domain sharing an edge and forming a convex quadrilateral, it is shown how to construct a non-negative basis for C1 splines that restrict to polynomials of a total degree higher than one on each of the triangles. The representation may be seen as a generalization of the Bernstein–Bézier form of a spline on every separate triangle, and the main challenge in its development

Fictitious domain methods enable to solve physical problems on unfitted grids, thereby avoiding time-consuming and error prone meshing phases. However, an accurate integration of the weak formulation is still mandatory, leading to the need for efficient quadrature strategies in the elements that are partially located in the physical domain. Various methodologies have been proposed to this end. Among

In the framework of the development of new passive safety systems for the second and third generations of nuclear reactors, the numerical simulations, involving complex turbulent two-phase flows around thin or massive inflow obstacles, are privileged tools to model, optimize and assess new design shapes. In order to match industrial demands, computational fluid dynamics tools must be the fastest, most

Drug release from viscoelastic polymeric matrices is a complex phenomenon where the main actors are the fluid, the polymeric structure and the drug. As the fluid enters into the polymer, the polymeric chains relax inducing a resistance to the fluid entrance. In contact with the fluid, a dissolution processes takes place and the dissolved drug diffuses through the medium. Our main goal in this paper

SCB is an efficient fluid solver developed for computing two-phase compressible flows involving strong shocks and expansion waves. It solves a four-equation diffuse-interface model, which is derived from the five-equation model proposed by Kapila et al. The governing equations are discretized by a finite volume method with explicit time stepping. SCB uses a fully parallel environment via Message Passing

Nonconforming Morley finite element method is applied to a fourth order nonlinear reaction-diffusion problems. After deriving some regularity results to be used subsequently in our error analysis, Morley FEM is employed to discretize in the spatial direction to obtain a semidiscrete problem. A priori bounds for the discrete solution are derived and with the help of an auxiliary problem, optimal error

In classic Reduced Basis (RB) framework, we propose a new technique for the offline greedy error analysis which relies on a residual-based a posteriori error estimator. This approach is as an alternative to classical a posteriori RB estimators, avoiding a discrete inf-sup lower bound estimate. We try to use less common ingredients of the RB framework to retrieve a better approximation of the RB error

This paper discusses the practical development of space-time boundary element methods for the wave equation in three spatial dimensions. The employed trial spaces stem from simplex meshes of the lateral boundary of the space-time cylinder. This approach conforms genuinely to the distinguished structure of the solution operators of the wave equation, so-called retarded potentials. Since the numerical

Steady-state two-dimensional lid-driven square cavity flows at high Reynolds numbers are solved in this communication using a velocity-based formulation developed in the context of the improved element-free Galerkin–reduced integration penalty method (IEFG–RIPM). The analyses based on the IEFG–RIPM are performed under a standard Galerkin weak-formulation, i.e. without the need of introducing streamline-upwind

In this article, a staggered discontinuous Galerkin (SDG) approximation on rectangular meshes for elliptic problems in two dimensions is constructed and analyzed. The optimal convergence results with respect to discrete L2 and H1 norms are theoretically proved. Some numerical evidences to verify the optimal convergence rates are presented. Several numerical examples to the elliptic singularly perturbed

An alternating direction implicit (ADI) difference method is adopted to solve the two-dimensional time-fractional diffusion equation with Dirichlet boundary condition whose solution has some weak singularity at initial time. L1 scheme on uniform mesh is used to discretize the temporal Caputo fractional derivative. Pointwise-in-time error estimate is given for the fully discrete ADI scheme, where the

We consider two parallel-in-time approaches applied to a (reaction) diffusion problem, possibly non-linear. In particular, we consider PFASST (Parallel Full Approximation Scheme in Space and Time) and space-time multigrid strategies. For both approaches, we start from an integral formulation of the continuous time dependent problem. Then, a collocation form for PFASST and a discontinuous Galerkin discretization

Our aim through this paper is to describe a Krylov based projection method in order to reduce high-order dynamical systems. We focus on differential algebraic equations (DAEs) of index-2 that arise from spatial discretization of Stokes equations. An efficient algorithm based on a projection technique onto an extended block Krylov subspace that appropriately allows us to construct a reduced order system

We derive the priori and a posteriori error estimates of the weak Galerkin finite element method with the Crank-Nicolson time discretization for the parabolic equation in this paper. The priori error estimates are deduced based on existing priori error results of the corresponding elliptic projection problem. For the a posteriori error estimates, the elliptic reconstruction technique is introduced

Due to corner stress singularities, free vibration analysis of thin sectorial plates with arbitrary boundary supports including a free vertex is a rather challenging problem. In this paper, a novel rotation-free quadrature element formulation based on Kirchhoff plate theory is presented to solve the problem. Different from all existing quadrature thin plate element formulations, rotational degrees

This paper focuses on the utilization of local radial basis functions (LRBFs) based level set method (LSM) for topology optimization of two-dimensional thermal problems using both concentrated as well as uniformly distributed heat generation. The design domain is embedded implicitly into a higher-dimensional function, which is parametrized with the LRBFs through an explicit scheme. This novel combination

We introduce a new symmetric treatment of anisotropic viscous terms in the viscoelastic wave equation. An appropriate memory variable treatment of stress-strain convolution terms, result into a symmetric system of first order linear hyperbolic partial differential equations, which we discretize using a high-order discontinuous Galerkin finite element method. The accuracy of the resulting numerical

The moving particle semi-implicit method (MPS) is a well-known Lagrange method that offers advantageous in addressing complex fluid problems, but particle distribution is an area that requires refinement. For this study, a particle smoothing algorithm was developed and incorporated into the weakly compressible MPS (sWC-MPS). From the definition and derivation of basic MPS operators, uniform particle

The novelty of the current study is the development of an enhanced finite shell model using the first order shear deformation theory (FSDT) to study the vibration characteristics of imperfect functionally graded (FGM) plates and shells including porosities. Unlike standard (FSDT) theory, the proposed model presents an improvement of the FSDT via the introduction of a quadratic shear function allowing

Let Γ be a smooth curve inside a two-dimensional rectangular region Ω. In this paper, we consider the Poisson interface problem −∇2u=f in Ω∖Γ with Dirichlet boundary condition such that f is smooth in Ω∖Γ and the jump functions [u] and [∇u⋅n→] across Γ are smooth along Γ. This Poisson interface problem includes the weak solution of −∇2u=f+gδΓ in Ω as a special case. Because the source term f is possibly

The conforming finite element Galerkin method is applied to discretise in the spatial direction for a class of strongly nonlinear parabolic problems. Using elliptic projection of the associated linearised stationary problem with Gronwall type result, optimal error estimates are derived, when piecewise polynomials of degree r≥1 are used, which improve upon earlier results of Axelsson ((1977) [3]) requiring

In this study, two schemes to treat fluid-solid interaction on complex boundaries are proposed and solved in the framework of lattice Boltzmann method. The schemes are applied on the phase-field-based wettability implementation methods for binary and ternary fluids to simulate static contact angle on a circular surface generated by the staircase approximation. For the binary system, surface-energy

This work gives a general framework for the convergence analysis of low-order nonconforming elements for the 3D Brinkman model over cubical meshes, including the theory of uniform convergence with respect to the given parameters and the high accuracy analysis in the Darcy region. We apply our theory to three elements known in other literature and a newly constructed edge-face-based element to show

Numerical methods for Navier-Stokes equation and elasticity problem lead to a class of 2×2 block structure linear system. Based on a general splitting for the (1, 1) block of coefficient matrix and the Uzawa method, we consider a general Uzawa-type method for solving this 2×2 block structure linear system in this paper. With different choices of splitting matrix, the proposed Uzawa-type method includes

Peridynamics is a nonlocal theory for dynamic fracture analysis consisting in a second order in time partial integro-differential equation. In this paper, we consider a nonlinear model of peridynamics in a two-dimensional spatial domain. We implement a spectral method for the space discretization based on the Fourier expansion of the solution while we consider the Newmark-β method for the time marching

The focus of this paper is on a implicit Backward-Euler (BE) scheme with the mixed finite element method (FEM) for the two-dimentional general Rosenau-RLW equation. In which the bilinear element is used to approximate the exact solution v and the variable p=−Δv, and the zero-order nédéléc's finite element to the variable q→=ϕ∇v, respectively. An important point is that with the proposed scheme we are

We consider an elliptic partial differential equation with a random diffusion parameter discretized by a stochastic collocation method in the parameter domain and a finite element method in the spatial domain. We prove for the first time convergence of a stochastic collocation algorithm which adaptively enriches the parameter space as well as refines the finite element meshes.

This work focuses on the free vibration characteristics of the functionally graded piezoelectric (FGPE) plate with classical and elastic constraints. First, the analytical model is established for the FGPE plate on the basis of the first-order shear deformation theory (FSDT). Besides, different distribution profiles of the material constituent are considered for the FGPE plate model, and various external

The goal of this paper is to introduce and study a new class of fractional differential hemivariational inequality formulated by an evolutionary hemivariational inequality and a fractional differential equation in Banach spaces. By employing the Rothe method and the surjectivity result, we derive the existence of unique solution for such a problem under some mild conditions. Moreover, we use the fully

In this paper, we establish a reduced-order finite element (ROFE) method with very few degrees of freedom for the incompressible miscible displacement problem. Firstly, we construct the finite element (FE) method with second-order accuracy in time, where backward difference formulation is used for the time discretization and classical FE formulation is used for the space discretization. Optimal a priori

We consider a two-phase Darcy flow in a fractured porous medium consisting in a matrix flow coupled with a tangential flow in the fractures, described as a network of planar surfaces. This flow model is also coupled with the mechanical deformation of the matrix assuming that the fractures are open and filled by the fluids, as well as small deformations and a linear elastic constitutive law. The model

In this paper, a new preconditioned iterative method is presented to solve a class of nonsymmetric nonsingular or singular saddle point problems. The implementation of the proposed preconditioned Krylov subspace method avoids solving inverse of Schur complement and only needs to solve one linear sub-system at each step, which implies that it may save considerable costs. Theoretical convergence analysis

A new generalised two-dimensional time and space variable-order fractional Bloch-Torrey equation is developed in this study. The variable-order Riesz fractional derivative and variable diffusion coefficient are introduced to simulate diffusion phenomena in heterogeneous, irregularly shaped biological tissues. The fractional Bloch-Torrey equation is discretised by the weighted and shifted Grünwald-Letnikov

In this paper, a three-dimensional time-dependent Riesz space-fractional diffusion equation is considered, and an alternating direction implicit (ADI) difference scheme is proposed, in which a weighted and shifted Grünwald difference scheme (see Tian et al. (2015) [33]) is utilized for the discretizations of space-fractional derivatives, and a fractional-Douglas-Gunn type ADI method is utilized for

In this paper we intend to construct a structure preserving difference scheme for two-dimensional space-fractional nonlinear Schrödinger (2D SFNS) equation with the integral fractional Laplacian. The temporal direction is discretized by the modified Crank-Nicolson method, and the spatial variable is approximated by a novel fractional central difference method. The mass and energy conservations and

This paper presents the numerical analysis for a variant of the nonlinear multiscale Dynamic Diffusion (DD) method for the advection-diffusion-reaction equation initially proposed by Arruda et al. [1] and recently studied by Valli et al. [2]. The new DD method, based on a two-scale approach, introduces locally and dynamically an extra stability through a nonlinear operator acting in all scales of the

In this paper we introduce a new method called the Dirac Assisted Tree (DAT) method, which can handle 1D heterogeneous Helmholtz equations with arbitrarily large variable wave numbers. DAT breaks an original global problem into many parallel tree-structured small local problems, which are linked together to form a global solution by solving small linking problems. To solve the local problems in DAT

A time-fractional diffusion problem is considered on a spherically symmetric domain in Rd for d=1,2,3. The solution of such a problem is shown in general to have a weak singularity near the initial time t=0, and bounds on certain derivatives of this solution are derived. To solve the problem numerically, spatial polar coordinates are used; a finite difference method is constructed on a mesh that is

In this paper, adaptive direct discontinuous Galerkin method for second order elliptic equations is studied in two dimensional. A numerical flux with general weighted averages and proper weights for interface problems is introduced. In addition to the optimal a priori error estimator, we propose a residual-type a posteriori error estimator. The global upper bound and local lower bound for the error

In this paper, nonlinear parabolic partial differential equations are considered to approximate by multiscale mortar mixed method. The key idea of the multiscale mortar mixed approach is to decompose the domain into the smaller subregions separated by the interfaces with the Dirichlet pressure boundary condition. Each subdomain is partitioned independently on the fine scale and the standard mixed methods

This article presents a parallel algorithm for seismic imaging based on depth wavefield extrapolation (the solution of a one-way wave equation [OWE]) employing the pseudospectral method. The algorithm is essentially oriented on common-offset vector (COV) gathers seismic migration based on an OWE, includes several parallelism levels, and utilizes message passing interface (MPI), Nested OpenMP, and CUDA