Predictor–corrector <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml128" display="inline" overflow="scroll" altimg="si90.gif"><mml:mi>p</mml:mi></mml:math>- and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml129" display="inline" overflow="scroll" altimg="si129.gif"><mml:mi>h</mml:mi><mml:mi>p</mml:mi></mml:math>-versions of the finite element method for Poisson’s equation in polygonal domains

Math Methods in App Sciences

Jung

2023

Solutions of boundary value problems for linear partial differential equations are known to exhibit singular behaviors near the boundary of nonsmooth domains. In this case, one is usually interested on the asymptotic behavior of the solutions near the geometric singularities. However, for both mathematical and engineering purposes, it is important to have extraction formulas for the coefficients in the asymptotic expansion. In this paper, we consider initial boundary value problems for the wave equation in two‐dimensional domains with corners and derive, by means of Fourier analysis, explicit computable expressions for the coefficients in the asymptotic expansion of the solution near the corners. The explicit structure of the singular solutions presented herein makes it fairly easy to compute numerically. Thus, the formulas can be used to construct postprocessing procedures for improving the accuracy of standard numerical techniques of approximating the solution and for improving the rates of convergence in the error estimates. The analysis presented herein would also apply to more general hyperbolic or parabolic equations with constant coefficients.

“…In real computations of the coefficients

{c}_n

{c}_n&#x0005E;K(t)&#x0003D;\frac{1}{2&#x0005E;{\mu_n}\Gamma \left({\mu}_n&#x0002B;1\right)}\sum \limits_{k&#x0003D;1}&#x0005E;K{\lambda}_{n,k}&#x0005E;{\mu_n}{u}_{n,k}(t),\kern0.30em t&gt;0.

…”

Section: The Model Problem In a Circular Sectormentioning

confidence: 99%

The coefficients in the asymptotic expansion of solutions of second‐order hyperbolic problems in polygonal domains

Math Methods in App Sciences

Jung

2023

“…Our algorithm makes use of the extraction formulas for the coefficients of the singularities, and it consists of starting with some initial values of the coefficients and then computing the coefficients and the regular part of the solution iteratively, hence the name "Predictor-corrector finite element method," refer [23] for the corresponding algorithm for Poisson's equation on polygonal domains. e present algorithm provides several advantages.…”

Section: Introductionmentioning

confidence: 99%

A Predictor-Corrector Finite Element Method for Time-Harmonic Maxwell’s Equations in Polygonal Domains

Mathematical Problems in Engineering

Nkeck

2020

The overall efficiency and accuracy of standard finite element methods may be severely reduced if the solution of the boundary value problem entails singularities. In the particular case of time-harmonic Maxwell’s equations in nonconvex polygonal domains Ω, H1-conforming nodal finite element methods may even fail to converge to the physical solution. In this paper, we present a new nodal finite element adaptation for solving time-harmonic Maxwell’s equations with perfectly conducting electric boundary condition in general polygonal domains. The originality of the present algorithm lies in the use of explicit extraction formulas for the coefficients of the singularities to define an iterative procedure for the improvement of the finite element solutions. A priori error estimates in the energy norm and in the L2 norm show that the new algorithm exhibits the same convergence properties as it is known for problems with regular solutions in the Sobolev space H2Ω2 in convex and nonconvex domains without the use of graded mesh refinements or any other modification of the bilinear form or the finite element spaces. Numerical experiments that validate the theoretical results are presented.

“…This method has been shown to be very efficient, especially in combination with the p-and hp-versions of the finite element method as in these cases the finite element solutions exhibit exponential convergence (cf. [26]).…”

Section: Introductionmentioning

confidence: 99%

“…Thus, the coefficients can be approximated once an approximate solution of the boundary value problem is available (cf. [24, 26]). This method is similar to the dual singular function method.…”

Section: Introductionmentioning

confidence: 99%

“…Since the algorithm consists of computing an initial finite element solution of the boundary value problem and then correcting it iteratively, we call it a predictor–corrector finite element method , see [26] in the case of Poisson's equation in polygonal domains and [24, 25] for Poisson equation in three‐dimensional domains with rotationally symmetric edges.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Singular solutions of the Poisson equation at edges of three‐dimensional domains and their treatment with a predictor–corrector finite element method

Numerical Methods Partial

Jung

2020

Solutions of boundary value problems in three-dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ⊂ R 3 with edge singularities and presents, on the one hand, explicit computational formulas for the flux intensity functions. On the other hand, it proposes and analyzes a nonconforming finite element method on regular meshes for the efficient treatment of the singularities. The novelty of the present method is the use of the explicit formulas for the flux intensity functions in defining a postprocessing procedure in the finite element approximation of the solution. A priori error estimates in H 1 (Ω) show that the present algorithm exhibits the same rate of convergence as it is known for problems with regular solutions.