The backward euler anisotropic a posteriori error analysis for parabolic integro-differential equations

Although two-dimensional (2D) parabolic integro-differential equations (PIDEs) arise in many physical contexts, there is no generally available software that is able to solve them numerically. To remedy this situation, in this article, we provide a compact implementation for solving 2D PIDEs using the finite element method (FEM) on unstructured grids. Piecewise linear finite element spaces on triangles are used for the space discretization, whereas the time discretization is based on the backward-Euler and the Crank–Nicolson methods. The quadrature rules for discretizing the Volterra integral term are chosen so as to be consistent with the time-stepping schemes; a more efficient version of the implementation that uses a vectorization technique in the assembly process is also presented. The compactness of the approach is demonstrated using the software Matrix Laboratory (MATLAB). The efficiency is demonstrated via a numerical example on an L-shaped domain, for which a comparison is possible against the commercially available finite element software COMSOL Multiphysics. Moreover, further consideration indicates that COMSOL Multiphysics cannot be directly applied to 2D PIDEs containing more complex kernels in the Volterra integral term, whereas our method can. Consequently, the subroutines we present constitute a valuable open and validated resource for solving more general 2D PIDEs.

SummaryWe present an algorithm to simulate curvilinear hydraulic fractures in plane strain and axisymmetry. We restrict our attention to sharp fractures propagating in an isotropic, linear elastic medium and driven by the injection of a laminar, Newtonian fluid governed by lubrication theory, and we require the existence of a finite lag region between the fluid front and the crack tip. The key novelty of our approach is in how we discretize the evolving crack and fluid domains: we utilize universal meshes (UMs), a technique to create conforming triangulations of a problem domain by only perturbing nodes of a universal background mesh in the vicinity of the boundary. In this way, we construct meshes, which conform to the crack and to the fluid front. This allows us to build standard piecewise linear finite element spaces and to monolithically solve the quasistatic hydraulic fracture problem for the displacement field in the rock and the pressure in the fluid. We demonstrate the performance of our algorithms through three examples: a convergence study in plane strain, a comparison with experiments in axisymmetry, and a novel case of a fracture in a narrow pay zone.

We present a new numerical method for solving nonlinear reaction‐diffusion systems with cross‐diffusion which are often taken as mathematical models for many applications in the biological, physical, and chemical sciences. The two‐dimensional system is discretized by the local discontinuous Galerkin (LDG) method on unstructured triangular meshes associated with the piecewise linear finite element spaces, which can derive not only numerical solutions but also approximations for fluxes at the same time comparing with most of study work up to now which has derived numerical solutions only. Considering the stability requirement for the explicit scheme with strict time step restriction (), the implicit integration factor (IIF) method is employed for the temporal discretization so that the time step can be relaxed as Δt = O(hmin). Moreover, the method allows us to compute element by element and avoids solving a global system of nonlinear algebraic equations as the standard implicit schemes do, which can reduce the computational cost greatly. Numerical simulations about the system with exact solution and the Brusselator model, which is a theoretical model for a type of autocatalytic chemical reaction, are conducted to confirm the expected accuracy, efficiency, and advantages of the proposed schemes.

A monotone finite volume element scheme for diffusion equations on arbitrary polygonal grids

A new numerically stable sequential algorithm for coupled finite-strain elastoplastic geomechanics and flow

In this paper, adaptive algorithms for time and space discretizations are added to an existing solution method previously applied to the Venice Lagoon Tidal Circulation problem. An analysis of the interactions between space and time discretizations adaptation algorithms is presented. In particular, it turns out that both error estimations in space and time must be present for maintaining the adaptation efficiency. Several advantages, for adaptivity and for time decoupling of the equations, offered by the operator-splitting adopted for shallow water equations solution are presented. Copyright © 1999 John Wiley & Sons, Ltd.

This paper proposes virtual element methods for approximating the mathematical model consisting of coupled poroelastic and Advection-Diffusion-Reaction (ADR) equations. The space discretization relies on virtual element spaces containing piecewise linear polynomials as well as non-polynomials for displacement, pressure and concentrations, and piecewise constant for total pressure; a backwardEuler scheme is employed for the approximation of time derivative. Using standard techniques of explicit schemes, the well-posedness of the resultant fully discrete scheme is derived. Moreover, under certain regularity assumptions on the mesh, optimal apriori error estimates are established by introducing suitable projection operators. Several numerical experiments are presented to validate the theoretical convergence rate and exhibit the proposed scheme’s performance.

This paper studies a residual-based a posteriori error estimates for linear parabolic interface problems in a bounded convex polygonal domain in $$\mathbb {R}^2$$R2. We use the standard linear finite element spaces in space which are allowed to change in time and the two-step backward differentiation formula (BDF-2) approximation at equidistant time step is used for the time discretizations. The essential ingredients in the error analysis are the continuous piecewise quadratic space---time BDF-2 reconstruction and Scott---Zhang interpolation estimates. Optimal order in time and an almost optimal order in space error estimates are derived in the $$L^{\infty }(L^{2})$$Lź(L2)-norm using only energy method. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical experiments are performed to validate the asymptotic behaviour of the derived error estimators.

Singular source terms in sub-diffusion equations may lead to the unboundedness of solutions, which will bring a severe reduction of convergence order of existing time-stepping schemes. In this work, we propose two efficient time-stepping schemes for solving sub-diffusion equations with a class of source terms mildly singular in time. One discretization is based on the Gr{\"u}nwald-Letnikov and backward Euler methods. First-order error estimate with respect to time is rigorously established for singular source terms and nonsmooth initial data. The other scheme derived from the second-order backward differentiation formula (BDF) is proved to possess second-order accuracy in time. Further, piecewise linear finite element and lumped mass finite element discretizations in space are applied and analyzed rigorously. Numerical investigations confirm our theoretical results.

We introduce and analyze a post-processing for continuous variational space-time discretizations of wave problems. The post-processing lifts the fully discrete approximation in time from a continuous to a continuously differentiable one. Further, it increases the order of convergence. The discretization in time is based on the Gauss–Lobatto quadrature formula which is essential for ensuring the improved convergence behavior. Error estimates of optimal order in various norms are proved. A bound of superconvergence at the discrete time nodes is included. To show the error estimates, a special approach is developed. First, error estimates for the time derivative of the post-processed solution are proved. In a second step these results are used to show the error estimates for the post-processed solution itself. The need for this approach comes through the structure of the wave equation. Stability properties of its solution preclude us from using absorption arguments for the control of certain error quantities. A further key ingredient of this work is the construction of a time-interpolate of the exact solution that is needed in an essential way. Finally, a conservation of energy property is shown for the post-processed solution which is an important feature for approximation schemes to wave equations. The error estimates are confirmed by numerical experiments.

The article presents the results of computational and experimental studies of the distribution of a model material (plastic spherical particles with a size of 6 mm) along the height of a laboratory two-dimensional apparatus of the fluidized bed of the periodic principle of action. To experimentally determine the distribution of the solid phase over the height of the apparatus, digital photographs of the fluidized bed were taken, which were then analyzed using an algorithm that had been specially developed for this purpose. The algorithm involved splitting the image by height into separate rectangular areas, identifying the particles and counting their number in each of these areas. Numerical experiments were performed using the previously proposed one-dimensional cell model of the fluidization process, constructed on the basis of the mathematical apparatus of the theory of Markov chains with discrete space and time. The design scheme of the model assumes the spatial decomposition of the layer in height into individual elements of small finite sizes. Thus, the numerically obtained results qualitatively corresponded to the full-scale field experiment that had been set up. To ensure the quantitative reliability of the calculated forecasts, a parametric identification of the model was performed using known empirical dependencies to calculate the particle resistance coefficient and estimate the coefficient of their macrodiffusion. A comparison of the results of numerical and field experiments made us possible to identify the most productive empirical dependencies that correspond to the cellular scheme of modeling the process. The resulting physical and mathematical model has a high predictive efficiency and can be used for engineering calculations of devices with a fluidized bed, as well as for setting and solving problems of optimal control of technological processes in these devices for various target functions.

New investigations on the method of characteristics for the evaluation of line transients

We study the numerical approximation to a nonlocal Volterra integro‐differential equation, in which the integral term is the convolution product of a positive‐definite kernel and a nonlocal peridynamic differential operator (PDDO). Compared with the classical differential operators, the nonlocal PDDOs describe, for example, discontinuities and have demonstrated more widespread applications. The equation is discretized in space by the Galerkin finite element method, and we accordingly prove its error estimate. We then discretize the equation in time by the backward Euler method, and a positive quadrature rule is combined to approximate the convolution term. The convergence rate of the fully‐discrete finite element scheme is proved, and numerical experiments are carried out to substantiate the theoretical findings.

In this article, we study the time dependent convection–diffusion–reaction equation coupled with the Darcy equation. We propose and analyze two numerical schemes based on finite element methods for the discretization in space and the implicit Euler method for the discretization in time. An optimal a priori error estimate is then derived for each numerical scheme. Finally, we present some numerical experiments that confirm the theoretical accuracy of the discretization.

In this paper the question of asymptotic stability for retarded functional reaction diffusion equations is faced. Due to the infinite dimension of the problem a numerical approach is necessary. Here we propose a technique based on a pseudospectral discretization in time and on a spectral discretization in space of the infinitesimal generator associated to the semigroup of solution operators. Some numerical experiments on a Hutchinson-type equation modeling heat conduction in a rod with spatially variable gain delayed feedback are performed to show the efficiency of the scheme. This work represents a nontrivial extension of previous work of the authors on the computation of asymptotic stability for delay differential equations.