Numerical Mathematics

The main goal of this paper is to establish the convergence of mimetic discretizations of the first-order system that describes linear diffusion. Specifically, mimetic discretizations based on the support-operators methodology (SO) have been applied successfully in a number of application areas, including diffusion and electromagnetics. These discretizations have demonstrated excellent robustness, however, a rigorous convergence proof has been lacking. In this research, we prove convergence of the SO discretization for linear diffusion by first developing a connection of this mimetic discretization with Mixed Finite Element (MFE) methods. This connection facilitates the application of existing tools and error estimates from the finite element literature to establish convergence for the SO discretization. The convergence properties of the SO discretization are verified with numerical examples.

This article focusses on the analysis of a conforming finite element method for the time-dependent incompressible Navier–Stokes equations. For divergence-free approximations, in a semi-discrete formulation, we prove error estimates for the velocity that hold independently of both pressure and Reynolds number. Here, a key aspect is the use of the discrete Stokes projection for the error splitting. Optionally, edge-stabilisation can be included in the case of dominant convection. Emphasising the importance of conservation properties, the theoretical results are complemented with numerical simulations of vortex dynamics and laminar boundary layer flows.

Nonlinear equations /systems appear in most science and engineering models. For example, when solving eigen value problems, optimization problems, differential equations, in circuit analysis, analysis of state equations for a real gas, in mechanical motions /oscillations, weather forecasting, integral equations, image processing and many other fields of engineering designing processes. Nonlinear systems /problems are difficult to solve manually but they occur naturally in fluid motions, heat transfer, wave motions, chemical reactions, etc. This study deals with construction of iterative methods for nonlinear root finding, applying Taylor’s series approximation of a nonlinear function f(x) combined with a new correction term in a quadratic or cubic model. Competent iterative algorithms of higher order were investigated. For test of convergence and efficiency, we applied basic theorems and solved some equations in C++. Keywords – nonlinear equations, Taylor’s approximation, iterative ...

The reduced basis (RB) method is proposed for the approximation of multiparametrized equations governing an optimal control problem. The idea behind the RB method is to project the solution onto a space of small dimension, specifically designed on the problem at hand, and to decouple the generation and projection stages (off-line/on-line computational procedures) of the approximation process in order to solve parametrized equations in a rapid and not expensive way. The application that we investigate is an air ...

In this work we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable e.g. in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profile...

We consider a conforming finite element approximation of the Reissner-Mindlin eigenvalue system, for which a robust a posteriori error estimator for the eigenvector and the eigenvalue errors is proposed. For that purpose, we first perform a robust a priori error analysis without strong regularity assumption. Upper and lower bounds are then obtained up to higher order terms that are superconvergent, provided that the eigenvalue is simple. The convergence rate of the proposed estimator is confirmed by a numerical test.

We present a higher order generalization for relaxation methods in the framework presented by Jin and Xin in . The schemes employ general higher order integration for spatial discretization and higher order implicit-explicit (IMEX) schemes or Total Variation diminishing (TVD) Runge-Kutta schemes for time integration of relaxing or relaxed schemes, respectively, for time integration. Numerical experiments are performed on various test problems, in particular, the Burger's and Euler equations of inviscid gas dynamics in both one and two space dimensions. In addition, uniform convergence with respect to the relaxation parameter is demonstrated.

In this work we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable e.g. in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profile...

Abstract—We consider the Stokes problem on a plane polygonal domain Ω⊂ R2. We propose a finite element method for overlapping or nonmatching grids for the Stokes problem based on the partition of unity method. We prove that the discrete inf-sup condition holds with a constant independent of the overlapping size of the subdomains. The results are valid for multiple subdomains and any spatial dimension. Keywords: non-matching grid, finite element method, partition of unity, Stokes problem

In this paper we introduce and analyze a new approach for the numerical approximation of Maxwell's equations in the frequency domain. Our method belongs to the recently proposed family of negative-norm least-squares algorithms for electromagnetic problems which have already been applied to the electrostatic and magnetostatic problems as well as the Maxwell eigenvalue problem (see ). The scheme is based on a natural weak variational formulation and does not employ potentials or "gauge conditions". The discretization involves only simple, piecewise polynomial, finite element spaces, avoiding the use of the complicated Nédélec elements. An interesting feature of this approach is that it leads to simultaneous approximation of the magnetic and electric fields, in contrast to other methods where one of the unknowns is eliminated and is later computed by differentiation. More importantly, the resulting discrete linear system is well-conditioned, symmetric and positive definite. We demonstrate that the overall numerical algorithm can be efficiently implemented and has an optimal convergence rate, even for problems with low regularity.

Multiscale radiative heat transfer (RHT) problems are formulated and methods to approximate their numerical solutions are developed. We focus on RHT problems in participating media with heteregeneous optical properties leading to both optically thick and thin regimes within the same media spectrum. By introducing a diffusive scale and using an assymtotic expansion in the RHT equations we formulate the simplified P N approximations. The optical spectrum is decomposed in wavelenght bands and the RHT equations are solved for bands with low absorption while the simplified P N equations are solved for bands with high absorption. The hybrid models solve the multiscale RHT more accuartely than the simplified P N approximations and with a computational cost less than using the full RHT solver. Accuracy and effectivness of the proposed models are demonstrated on three-dimensional RHT problems arising in combustion systems.

In this paper, we consider variational inequalities related to problems with nonlinear boundary conditions. We are focused on deriving a posteriori estimates of the difference between exact solutions of such type variational inequalities and any function lying in the admissible functional class of the problem considered. These estimates are obtained by an advanced version of the variational approach earlier used for problems with uniformly convex functionals (see [13,). It is shown that the structure of error majorants reflects properties of the exact solution. The majorants provide guaranteed upper bounds of the error for any conforming approximation and possess necessary continuity properties. In the series of numerical tests performed, it was shown that the estimates are explicitly computable, provide sharp bounds of approximation errors, and give high quality indication of the distribution of local (elementwise) errors.

This article focusses on the analysis of a conforming finite element method for the time-dependent incompressible Navier-Stokes equations. For divergence-free approximations, in a semi-discrete formulation, we prove error estimates for the velocity that hold independently of both pressure and Reynolds number. Here, a key aspect is the use of the discrete Stokes projection for the error splitting. Optionally, edge-stabilisation can be included in the case of dominant convection. Emphasising the importance of conservation properties, the theoretical results are complemented with numerical simulations of vortex dynamics and laminar boundary layer flows.

Bounds on the spectrum of Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients of the convergence analysis of FETI (finite element tearing and interconnecting) based domain decomposition methods. Here we give bounds on the regular condition number of Schur complements of ‘floating’ clusters arising from the discretization of 3D Laplacian on a cube decomposed into cube subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m × m cube subdomains connected by face and optionally edge averages increases proportionally to m. The estimates support scalability of unpreconditioned H-FETI-DP (hybrid FETI dual-primal) method. Though the research is most important for the solution of variational inequalities, the results of numerical experiments indicate that unpreconditioned H-FETI-DP with large clusters can be useful also for the solution of huge linear problems.

In this paper, we deal with the discontinuous Galerkin finite element method for the plate contact problem with the frictional boundary conditions. The weak formulation is the variational inequality problem of the second kind. In virtue of the special discrete variational formulation, the error estimate in the broken H 2 norm is derived between the discontinuous approximation solution and the exact solution.

The method of dynamic relaxation in its early stages of development was
perceived as a numerical finite difference technique. It was first used to analyze
structures, then skeletal and cable structures, and plates. The method relies on a
discretized continuum in which the mass of the structure is assumed to be
concentrated at given points (i.e. nodes) on the surface. The system of concentrated
masses oscillates about the equilibrium position under the influence of out of balance
forces. With time, it comes to rest under the influence of damping. The iterative
scheme reflects a process, in which static equilibrium of the system is achieved by
simulating a pseudo dynamic process in time. In its original form, the method makes
use of inertia term, damping term and time increment.
Chapter one includes introduction to dynamic relaxation method (DR) which is
combined with finite differences method (FD) for the sake of solving ordinary and
partial differential equations, as a single equation or as a group of differential
equations. In this chapter the dynamic relaxation equations are transformed to
artificial dynamic space by adding damping and inertia effects. These are then
expressed in finite difference form and the solution is obtained through iterations.
In chapter two the procedural steps in solving differential equations using DR
method were applied to the system of differential equations (i.e. ordinary and/ or
partial differential equations). The DR program performs the following operations:
Reads data file; computes fictitious densities; computes velocities and displacements;
checks stability of numerical computations; checks convergence of solution; and
checks wrong convergence. At the end of this chapter the dynamic relaxation (DR)
numerical method coupled with the finite differences discretization technique is used
to solve nonlinear ordinary and partial differential equations. Subsequently, a
FORTRAN program is developed to generate the numerical results as analytical and/
or exact solutions.
ix
Chapter three discusses the importance of using numerical methods in solving
differential equations and stresses the use of dynamic relaxation technique in these
applications.
The book is suitable as a textbook for a first course on dynamic relaxation
technique in civil and mechanical engineering curricula. It can be used as a reference
by engineers and scientists working in industry and academic institutions.
Author
Assistant Professor
Osama Mohammed Elmardi Suleiman

In this paper, we consider the degenerated isotropic boundary value problem −∇(ω 2 (x)∇u(x, y)) = f (x, y) on the unit square (0, 1) 2 . The weight function is assumed to be of the form ω 2 (ξ) = ξ α , where α ≥ 0. This problem is discretized by piecewise linear finite elements on a triangular mesh of isosceles right-angled triangles. The system of linear algebraic equations is solved by a preconditioned gradient method using a domain decomposition preconditioner with overlap. Two different preconditioners are presented and the optimality of the condition number for the preconditioned system is proved for α = 1. The preoconditioning operation requires O(N ) operations, where N is the number of unknowns. Several numerical experiments show the preformance of the proposed method.

In this paper, we consider the degenerated isotropic boundary value problem −∇(ω 2 (x)∇u(x, y)) = f (x, y) on the unit square (0, 1) 2 . The weight function is assumed to be of the form ω 2 (ξ) = ξ α , where α ≥ 0. This problem is discretized by piecewise linear finite elements on a triangular mesh of isosceles right-angled triangles. The system of linear algebraic equations is solved by a preconditioned gradient method using a domain decomposition preconditioner with overlap. Two different preconditioners are presented and the optimality of the condition number for the preconditioned system is proved for α = 1. The preoconditioning operation requires O(N ) operations, where N is the number of unknowns. Several numerical experiments show the preformance of the proposed method.

In this paper, we consider the degenerated isotropic boundary value problem −∇(ω 2 (x)∇u(x, y)) = f (x, y) on the unit square (0, 1) 2 . The weight function is assumed to be of the form ω 2 (ξ) = ξ α , where α ≥ 0. This problem is discretized by piecewise linear finite elements on a triangular mesh of isosceles right-angled triangles. The system of linear algebraic equations is solved by a preconditioned gradient method using a domain decomposition preconditioner with overlap. Two different preconditioners are presented and the optimality of the condition number for the preconditioned system is proved for α = 1. The preoconditioning operation requires O(N ) operations, where N is the number of unknowns. Several numerical experiments show the preformance of the proposed method.

We present a higher order generalization for relaxation methods in the framework presented by Jin and Xin in . The schemes employ general higher order integration for spatial discretization and higher order implicit-explicit (IMEX) schemes or Total Variation diminishing (TVD) Runge-Kutta schemes for time integration of relaxing or relaxed schemes, respectively, for time integration. Numerical experiments are performed on various test problems, in particular, the Burger's and Euler equations of inviscid gas dynamics in both one and two space dimensions. In addition, uniform convergence with respect to the relaxation parameter is demonstrated.

In many practical applications, for instance, in computational electromagnetics, the excitation is time-harmonic. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple elliptic equation for the amplitude. This is true for linear problems, but not for nonlinear problems. However, due to the periodicity of the solution,

The goal of this work is the presentation of some new formats which are useful for the approximation of (large and dense) matrices related to certain classes of functions and nonlocal (integral, integrodifferential) operators, especially for high-dimensional problems. These new formats elaborate on a sum of few terms of Kronecker products of smaller-sized matrices (cf. ). In addition to this we need that the Kronecker factors possess a certain data-sparse structure. Depending on the construction of the Kronecker factors we are led to so-called "profile-low-rank matrices" or hierarchical matrices (cf. ). We give a proof for the existence of such formats and expound a gainful combination of the Kronecker-tensor-product structure and the arithmetic for hierarchical matrices.

The purpose of this paper is to explain the phenomenon of symmetry breaking for optimal functions in functional inequalities by the numerical computations of some well chosen solutions of the corresponding Euler-Lagrange equations. For many of those inequalities it was believed that the only source of symmetry breaking would be the instability of the symmetric optimizer in the class of all admissible functions. But recently, it was shown by an indirect argument that for some Caffarelli-Kohn-Nirenberg inequalities this conjecture was not true. In order to understand this new symmetry breaking mechanism we have computed the branch of minimal solutions for a simple problem. A reparametrization of this branch allows us to build a scenario for the new phenomenon of symmetry breaking. The computations have been performed using Freefem++.

This work is devoted to the computation and study of properties of the mean quadratic fluctuation of energy in some quantum mechanical systems (multielectron atom, molecule, quarkonium in mechanical approximation) in the state described by an approximate wave function. The motivation for such a study is the investigation of a convenient "measure of approximation" of stationary Schroedinger equation. The functional (4) of "integral error" is defined for such purposes. It is identical to the mean quadratic energy fluctuation functional (6) which is not only a convenient measure of an error but it enables also the constructions of upper and lower bounds for the exact value of the energy E_n (in (1), chapter 2.1). I included also a short overview of the theory of orthogonal polynomials and some identities for Fourier transforms of distributions, needed in the computation of matrix elements of Hamiltonian square operator. The Generalized Slater-Condon rules are introduced and investigated in this work (tables Y1,Y2,Y3). It turns out that for multielectron atoms, the integrals appearing in the matrix elements of Hamiltonian square operator are analytical in the bases of Slater (STO), Gauss (GTO) and hydrogen (VTO (Czech Abbrev., Eng. should be HTO)) types. For a general molecule, an analytic expression of all (except of the 5-centered and 6-centered) integrals needed for the computation is found.

In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method was introduced which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. This method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. In the paper [C. Lehrenfeld and A. Reusken, IMA J. Numer. Anal. 38 (2018), No. 3, 1351–1387] an a priori error analysis of the method applied to an interface problem is presented. The analysis reveals optimal order discretization error bounds in the H1-norm. In this paper we extend this analysis and derive optimal L2-error bounds.

In the paper, Pythagorean-hodograph cycloidal curves as an extension of PH cubics are introduced. Their properties are examined and a constructive geometric characterization is established. Further, PHC curves are applied in the Hermite interpolation, with closed form solutions been determined. The asymptotic approximation order analysis carried out indicates clearly which interpolatory curve solution should be selected in practice. This makes the curves introduced here a useful practical tool, in particular in algorithms that guide CNC machines.

This is the fth paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial dierential system which describes the ow of two incompressible, immiscible uids in porous media. In this paper we study a nite element procedure for numerically solving this system. This procedure uses a mixed nite element method for the elliptic equation of the pressure and velocity and the modi ed method of characteristics for the degenerate parabolic equation of the saturation. Sharp error estimates in energy norms are obtained for this procedure. The error analysis does not use any regularization of the saturation equation and does not impose any restriction on the nature of degeneracy; these error estimates are derived directly from this degenerate saturation equation. Other characteristic nite element methods such as the modi ed method of characteristics with adjusted advection (MMOCAA) are also considered.

A general setting for a new approach to constructing vector splitting schemes in mixed FEM for heat transfer problem is considered. For space approximation Raviart–Thomas finite elements of lowest order are implemented on rectangular (parallelepiped) mesh. The main question discussed is how to implement time discretization so as to obtain efficient numerical algorithms.The key idea of the proposed approach is to use scalar splitting schemes for heat flux divergence. This allows one to carry out accuracy and stability analysis on the basis of the well-known results for underlying scalar splitting schemes.

... Sub-Scales Method Tomás Chacón Rebollo, Macarena Gómez Mármol, and Isabel Sánchez MuQnoz ... In this paper we analyze how to apply the OSS modeling to the linearized Prim-itive Equations, and propose a model. We next analyze the stability and accuracy of this model. ...

... Sub-Scales Method Tomás Chacón Rebollo, Macarena Gómez Mármol, and Isabel Sánchez MuQnoz ... In this paper we analyze how to apply the OSS modeling to the linearized Prim-itive Equations, and propose a model. We next analyze the stability and accuracy of this model. ...

We consider the Stokes Problem on a plane polygonal domain Ω ⊂ R 2 . We propose a finite element method for overlapping or nonmatching grids for the Stokes Problem based on the partition of unity method. We prove that the discrete inf-sup condition holds with a constant independent of the overlapping size of the subdomains. The results are valid for multiple subdomains and any spatial dimension.

We present a higher order generalization for relaxation methods in the framework presented by Jin and Xin in . The schemes employ general higher order integration for spatial discretization and higher order implicit-explicit (IMEX) schemes or Total Variation diminishing (TVD) Runge-Kutta schemes for time integration of relaxing or relaxed schemes, respectively, for time integration. Numerical experiments are performed on various test problems, in particular, the Burger's and Euler equations of inviscid gas dynamics in both one and two space dimensions. In addition, uniform convergence with respect to the relaxation parameter is demonstrated.

A nonoverlapping domain decomposition approach with uniform and matching grids is used to define and compute the orthogonal spline collocation solution of the Dirichlet boundary value problem for Poisson's equation on a square partitioned into four squares. The collocation solution on four interfaces is computed using the preconditioned conjugate gradient method with the preconditioner defined in terms of interface preconditioners for the adjacent squares. The collocation solution on four squares is computed by a matrix decomposition method that uses fast Fourier transforms. With the number of preconditioned conjugate gradient iterations proportional to log 2 N , the total cost of the algorithm is O(N 2 log 2 N ), where the number of unknowns in the collocation solution is O(N 2 ). The approach presented in this paper, along with that in [6], generalizes to variable coefficient equations on rectangular polygons partitioned into many subrectangles and is well suited for parallel computation.

The problem of interpolation of scattered data on the unit sphere has many applications in geodesy and Earth science in which the sphere is taken as a model for the Earth. Spherical radial basis functions provide a convenient tool for constructing the interpolant. However, the underlying linear systems tend to be ill-conditioned. In this paper, we present an additive Schwarz preconditioner for accelerating the solution process. An estimate for the condition number of the preconditioned system will be discussed. Numerical experiments using MAGSAT satellite data will be presented.

We are concerned with an a posteriori error analysis of adaptive finite element approximations of boundary control problems for second order elliptic boundary value problems under bilateral bound constraints on the control which acts through a Neumann type boundary condition. In particular, the analysis of the errors in the state, the adjoint state, the control, and the adjoint control invokes an efficient and reliable residual-type a posteriori error estimator as well as data oscillations. The proof of the efficiency and reliability is done without any regularity assumption. Adaptive mesh refinement is realized on the basis of a bulk criterion. The performance of the adaptive finite element approximation is illustrated by a detailed documentation of numerical results for selected test problems.

In this paper, we consider the degenerated isotropic boundary value problem −∇(ω 2 (x)∇u(x, y)) = f (x, y) on the unit square (0, 1) 2 . The weight function is assumed to be of the form ω 2 (ξ) = ξ α , where α ≥ 0. This problem is discretized by piecewise linear finite elements on a triangular mesh of isosceles right-angled triangles. The system of linear algebraic equations is solved by a preconditioned gradient method using a domain decomposition preconditioner with overlap. Two different preconditioners are presented and the optimality of the condition number for the preconditioned system is proved for α = 1. The preoconditioning operation requires O(N ) operations, where N is the number of unknowns. Several numerical experiments show the preformance of the proposed method.

In the context of multicomponent flows, we are faced with PDE systems solutions combining waves whose speeds are several orders of magnitude apart. Of these waves, only the slow kinematic ones that represent transport phenomena are of concern to us. The fast acoustic ones, although uninteresting for the physical application considered, nevertheless impose a prohibitively small time-step (via the classical CFL restriction) if treated explicitly. This is why we propose to use a hybrid finite-volume scheme in which fast waves are handled by a linearized implicit formulation and slow waves remain explicitly solved. To further decrease the computational cost, mostly due to the complexity of nonlinear thermodynamical laws, we combine this method with a fully adaptive multiresolution scheme. At each time step, a multiscale analysis followed by the thresholding of small details enables us to discretize the solution over a time-varying adaptive grid, based on the smoothness of the relevant phenomenon. Particular attention is given to the extension of the reference scheme to non uniform grid and to the prediction strategy of the adaptive grid from one time-step to another. Finally, efficiency in terms of computing time requirements is studied in conjunction with the accuracy performances.

This paper is concerned with the analysis of adaptive multiscale techniques for the solution of a wide class of elliptic operator equations covering, in principle, singular integral as well as partial differential operators. The central objective is to derive reliable and efficient a-posteriori error estimators for Galerkin schemes which are based on stable multiscale bases. It is shown that the locality of corresponding multiresolution processes combined with certain norm equivalences involving weighted sequence norms of wavelet coefficients leads to adaptive space refinement strategies which are guaranteed to converge in a wide range of cases, again including operators of negative order.