A highly-accurate finite element method with exponentially compressed meshes for the solution of the Dirichlet problem of the generalized Helmholtz equation with corner singularities

TL;DR: In this paper, a conforming finite element method for the Dirichlet problem of the generalized Helmholtz equation on domains with re-entrant corners is presented, where the k − t h order Lagrange elements are used for discretization of the variational form of the problem.

About: This article is published in Journal of Computational and Applied Mathematics.The article was published on 2019-12-01 and is currently open access. It has received 3 citations till now. The article focuses on the topics: Dirichlet problem & Partial differential equation.

Summary

1 Introduction

• Singularities are often encountered in the solution of elliptic equations in two dimensions due to the non-smoothness of the boundary of the domain and the abrupt changes of the boundary conditions.
• For the implementation of the proposed method, the exponentially compressed meshes are transformed to square meshes in Log-Polar coordinates, hence there is no need to design a special mesh or to derive a new algorithm for the solution in the neighbourhood of the corners.
• In Section 3 the authors describe the proposed method for computing the highly-accurate approximate solution of the generalized Helmholtz equation with Dirichlet boundary conditions on polygons.
• Section 4 is devoted to the error analysis of the method.
• In Section 5 the authors present the solutions of a numerical example solved by the proposed method, and nally concluding remarks are given in Section 6.

3 Description of the Method

• For the solution of problem (24) in G , a nite element mesh is formed using triangular and curved elements.
• The solution is based on k th order Lagrange elements, which are C0continuous, Pk nite elements.
• The authors introduce the parameter h > 0; which denotes the largest side in any element of the mesh on G (in the case of a curved element this means the triangle with the same vertices), and let G h denote the nite element mesh formed on G .
• The corresponding approximation problem to the variational problem (4) is as follows:.
• As the solution might only have a weak singularity in the vicinity of the corners with interior angle j when 0 < j < 1, j 2 E; there is no need to construct exponentially compressed polar meshes near these corners when the required accuracy of the approximate solution is O(h) in the H1-norm.

4 Error Analysis

• The corresponding approximation problem is: Everywhere below the authors will denote constants which are independent of h by c; c0; c1; :::; generally using the same notation for di erent constants for simplicity.
• Let u be the solution of the variational problem (4) and uh be the solution of the corresponding approximation problem (24).
• For simplicity, the authors keep the same notation.
• Let hu denote the Vh-interpolant of u; and let ûh 2.
• The authors may choose ûh so that it has the same value at all interior nodes as the interpolant hu.

5 Numerical Example

• To test the e ectiveness of the method, a numerical example is computed in an L-shaped domain (see Figure 2), where the exact solution has a corner singularity at the vertex A1 with an interior angle 1 = 3 =2:.
• For the implementation of the method, linear and bilinear Lagrange elements were used so that an accuracy of O(h2) is obtained in the L2 relative error norm, where h is de ned as in Section 3.
• The expected solution to problem (41), (42), cannot be computed with arbitrary accuracy since it requires the summation of an in nite Bessel series.
• The value of 1 is chosen such that the number of element nodes are consistent on each successive grid.

6 Concluding Remarks

• A highly-accurate nite element method has been developed and justi ed for the approximate solution of the generalized Helmholtz equation with Dirichlet boundary conditions on polygons.
• The implementation of the method is straight forward since the polar meshes transform to square meshes in Log-Polar coordinates.
• For the implementation of the numerical example, the approximate solution has been based on linear Lagrange polynomials.
• Furthermore, the proposed method can be applied in domains whose 'non-singular' part has a curved boundary.
• The authors note that this will also re ect on the choice of j given in (21).

Figures

Figure 1: A polygonal domain described in Section 2

Table 1: The relative error and convergence rate of problem (41), (42) in the L2 Norm.

Figure 2: Exponentially compressed polar mesh on a polygon with one singular corner.

Table 2: The convergence rate of problem (41), (42) on successive grids.

Frequently asked questions

Q1. What have the authors contributed in "University of dundee a highly-accurate finite element method with exponentially compressed meshes for the solution of the dirichlet problem of the generalized helmholtz equation with corner singularities" ?

In this study, a highly-accurate, conforming nite element method is developed and justi ed for the solution of the Dirichlet problem of the generalized Helmholtz equation on domains with re-entrant corners. 

Q2. What future works have the authors mentioned in the paper "University of dundee a highly-accurate finite element method with exponentially compressed meshes for the solution of the dirichlet problem of the generalized helmholtz equation with corner singularities" ?

However it will be possible to extend the method by taking into account the change that will be required in the boundary condition ( 22 ), which is applied on the arti cial boundary of the grid. 

Q3. Why are singularities often encountered in the solution of elliptic equations in two dimensions?

Singularities are often encountered in the solution of elliptic equations in two dimensions due to the non-smoothness of the boundary of the domain and the abrupt changes of the boundary conditions. 

Q4. What is the advantage of exponentially compressed meshes?

The use of exponentially compressed meshes also has the advantage of much faster re nement compared to algebraically re ned meshes, leading to a nite-element system with smaller number of equations. 

Q5. How did Wu and Han deal with the problem?

Wu and Han [8] dealt with the problem by introducing an arti cial boundary and using the nite-element method in the domain away from the singularities. 

Q6. What is the boundary function of the curved triangle?

On the arti cial boundary #j j ; the Lagrange polynomialj ( j) = ( j j) j + j j 1j (22)is applied as the boundary function, where ; = j 1; j; denotes the value of the boundary function at a point on , whose distance from the vertex 

Q7. What is the function u1 that solves the boundary value problem?

An asymptotic series for the function u1 that satis es problem (10), (11) can also be expressed asu1(rj ; j) = 1X m=1 rmj (Am( j) ln rj +Bm( j)) : (13)The particular solution uP is the solution of the following boundary value problem:uP + a0uP = f on Sj ; (14)uP j j 1 = 0; uP j j = 0: (15)For the series solution of problem (14), (15), the function f is expanded in the formf = 1X i=1 fi( j)r i j :With the use of the eigenfunction expansion of Green's function, the solution of problem (14), (15) can be written asuP (rj ; j) = 2j Z Sj 1X n=1 'n( j)'n( 

Q8. what is the corresponding approximation problem to the variational problem?

The authors denote by Vh the nite element subspace of H 1 E ; and by V0h the restriction of Vh to theboundary 0: Let uh 2 V 0h be the interpolant of the boundary functions (2) on n~ ; and (22) on the arti cial boundary #j j ; j 2 E:The corresponding approximation problem to the variational problem (4) is as follows: 

Q9. what is the kth order of the boundary functions?

The authors let f'1; :::; 'neg denote the set of basis functions of the nite element space V 0 h , de ne the additional functions 'i; i = ne + 1; :::; ne + n ; by extending the space V 0 h to Vh; and select the xed coe cients ui, i = ne + 1; :::; ne + n ; so thatuh = ne+n X i=ne+1 ui'i (27)is the k th order V 0h interpolant of the boundary functions (2) on n~ ; function (22) on #j j ; j 2 E; and has the value zero at all remaining nodes. 

Q10. What is the error bound of the solution on the nite element mesh?

For the error bound of the solution on the nite element mesh G h; taking (3), (17) and (27) into account, by Corollary 5 in [24], interpolation theory and Theorem 4 in [20], for any integer k 1 the authors have the estimateku uhk1;G h chk: (29)Now, the authors analyze the error bound of the solution on the sector Thj ; j 2 E: Without loss of generality, let the radius of the sector rj0 = 1:The authors consider 

Q11. How accurate is the convergence rate on the nite element grid?

Since this is not a very accurate measure, the authors also present the convergence rate obtained by comparing the numerical solution attained in Th1 on successive grids, where the convergence rate on the grid with h = 2 m is de ned as ku2 (m+1) u2 mk0;Th1ku2 (m+1) u2 (m+2)k0;Th1 :The O(h2) accuracy corresponds to 22 for convergence rate.