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For the non-preconditioned Galerkin matrix of the hypersingular integral operator, the condition number grows with the number of elements as well as the quotient of the maximal and the minimal mesh-size. Therefore, reliable and effective numerical computations, in particular on adaptively refined meshes, require the development of appropriate preconditioners. We propose and analyze a local multilevel...
In this paper, a simple numerical method is proposed for solving singularly perturbed boundary layers problems exhibiting twin boundary layers. The method avoids the choice of fitted meshes. Firstly the original problem is transformed into a new boundary value problem whose solution does not change rapidly by a proper variable transformation; then the transformed problem is solved by using the reproducing...
As a model of the second order elliptic equation with non-trivial boundary conditions, we consider the Laplace equation with mixed Dirichlet and Neumann boundary conditions on convex polygonal domains. Our goal is to establish that finite element discrete harmonic functions with mixed Dirichlet and Neumann boundary conditions satisfy a weak (Agmon–Miranda) discrete maximum principle, and then prove...
In this paper, we present the explicit and computable formula of the structured backward errors of the generalized saddle point systems. Simple numerical examples show that the expressions are useful for testing the stability of practical algorithms.
We propose mixed finite element methods for the standard linear solid model in viscoelasticity and prove a priori error estimates. In our mixed formulation the governing equations of the problem become a symmetric hyperbolic system, so we can use standard techniques for a priori error estimates and time discretization. Numerical results illustrating our theoretical analysis are included.
In this work, we develop a new linearized implicit finite volume method for chemotaxis-growth models. First, we derive the scheme for a simplified chemotaxis model arising in embryology. The model consists of two coupled nonlinear PDEs: parabolic convection-diffusion equation with a logistic source term for the cell-density, and an elliptic reaction-diffusion equation for the chemical signal. The...
A class of Fredholm integral equations of the second kind, with respect to the exponential weight function $$w(x)=\exp (-(x^{-\alpha }+x^\beta ))$$ w ( x ) = exp ( - ( x - α + x β ) ) , $$\alpha >0$$ α > 0 , $$\beta >1$$ β > 1 , on $$(0,+\infty )$$ ( 0 , + ∞ ) , is considered. The kernel k(x, y) and the function g(x) in such kind of equations,...
The local convergence analysis of a parameter based iteration with Hölder continuous first derivative is studied for finding solutions of nonlinear equations in Banach spaces. It generalizes the local convergence analysis under Lipschitz continuous first derivative. The main contribution is to show the applicability to those problems for which Lipschitz condition fails without using higher order derivatives...
This paper establishes a numerical validation test for solutions of systems of absolute value equations based on the Poincaré–Miranda theorem. In this paper, the Moore–Kioustelidis theorem is generalized for nondifferential systems of absolute value equations. Numerical results are reported to show the efficiency of the new test method.
This paper studies the problem of approximating a function f in a Banach space $$\mathcal{X}$$ X from measurements $$l_j(f)$$ l j ( f ) , $$j=1,\ldots ,m$$ j = 1 , … , m , where the $$l_j$$ l j are linear functionals from $$\mathcal{X}^*$$ X ∗ . Quantitative results for such recovery problems require additional information about the sought after...
In this note we shall devise a variable-order continuous Galerkin time stepping method which is especially geared towards norm-preserving dynamical systems. In addition, we will provide an a posteriori estimate for the $$L^\infty $$ L ∞ -error.
We consider the problem of solving a rational matrix equation arising in the solution of G-networks. We propose and analyze two numerical methods: a fixed point iteration and the Newton–Raphson method. The fixed point iteration is shown to be globally convergent with linear convergence rate, while the Newton method is shown to have a local convergence, with quadratic convergence rate. Numerical experiments...
Fractional calculus is a powerful and effective tool for modelling nonlinear systems. In this paper, we introduce a class of new fractional derivative named general conformable fractional derivative (GCFD) to describe the physical world. The GCFD is generalized from the concept of conformable fractional derivative (CFD) proposed by Khalil. We point out that the term $$t^{1-\alpha }$$ t 1 -...
We analyze the discretization of the periodic initial value problem for Kuramoto–Sivashinsky type equations with Burgers nonlinearity by implicit–explicit backward difference formula (BDF) methods, establish stability and derive optimal order error estimates. We also study discretization in space by spectral methods.
We find square roots of a complex-valued matrix $$A_{3 \times 3}$$ A 3 × 3 using equation $$B^{2}=A$$ B 2 = A . The proposed method is faster than Higham’s method and provides up to 8 square roots with less relative residual and error.
We study the asymptotic behavior of harmonic interpolation of harmonic functions based on Radon projections when the chords coalesce to some points, a chord and a point. We show that the limit is the Lagrange or Taylor-type interpolation at coalescing points or chords.
The PageRank algorithm is one of the most commonly used techniques that determines the global importance of Web pages. In this paper, we present a preconditioned Arnoldi-Inout approach for the computation of Pagerank vector, which can take the advantage of both a new two-stage matrix splitting iteration and the Arnoldi process. The implementation and convergence of the new algorithm are discussed...
A nonmonotone trust-region method for the solution of nonlinear systems of equations with box constraints is considered. The method differs from existing trust-region methods both in using a new nonmonotonicity strategy in order to accept the current step and a new updating technique for the trust-region-radius. The overall method is shown to be globally convergent. Moreover, when combined with suitable...
B-matrices form a subclass of P-matrices for which error bounds for the linear complementarity problem are known. It is proved that a bound involved in such problems is asymptotically optimal. $$B_\pi ^R$$ B π R -matrices form a subclass of P-matrices containing B-matrices. For the $$B_\pi ^R$$ B π R -matrices, error bounds for the linear complementarity problem are obtained. We...
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