A new and simple implementation of the element-local L2-projected continuous finite element method
Introduction
In this paper we would like to continue to carry out some deep studies on the continuous finite element (C0-FE) method for the problems of static electromagnetic fields, including the electrostatic problem, Maxwell’s equations, and the Maxwell’s eigenvalue problems. These models are famous and important, in both theory and numerics.
As a typical problem, let us begin this topic with the electrostatic problem; the others are similar. The electrostatic problem in an isotropic homogeneous medium is described as the div-curl systemwith a suitable boundary condition, for example, on will be considered in this paper. Here is a two-dimensional bounded connected domain with Lipschitz continuous boundary, and f and g are given data in .
Till now, many numerical methods have been given and studied for (1) in lots of literatures. If the exact solution of (1) belongs to , it is natural to adopt the C0-FE method. The existing works [1], [2], [3] have pointed out that this method performs very well for the smooth H1-solution which means with in this paper. However, the exact solution of (1) may have lower regularity [4], i.e., it merely belongs to with a certain constant , and named as the non-H1 solution in this paper. In this situation, the C0-FE approximation may give a wrong or very bad convergence; please refer to the papers [5], [6], [7], [8]. In order to overcome this obstacle and obtain the satisfying approximation solution to the non-H1 solution, some nonstandard finite element approximation techniques using C0-FE space are presented in last decade, for example, the singular field method [9], the weighted formulation [10], the negative method [8], [11], and the first order system LL∗ method [7].
Recently, the element-local -projected stabilized C0-FE method are presented in [6], [12], which is referred to as the original EL2P method in this paper. Theory analysis and numerical experiments have shown this method works very well to solve the non-H1 solution for the problems of static electromagnetic fields. In this paper we would like to focus our eyesight on this original EL2P method, and make a deep study on the implementation of this method. We want to establish a new and simple method based on the original EL2P method, which is easy to understand and implement.
To state our idea clearly, let us review here the implementation of the original EL2P method in [6] to resolve problem (1). Let be the C0-FE space satisfying the boundary condition. The approximation solution are sought by the variational form for all ,Here, two element-local L2-projections, denoted by and respectively, are embedded in the standard variational form of this problem, in order to overcome the difficulty due to the large kernel space about the div and curl operators. This treatment leads towhere is the usual L2-inner product in . Furthermore, in order to remedy the instability resulted from the above two element-local L2-projections, the authors introduced two mesh-dependent stabilization terms and . For more information, please see [6].
The success of the original EL2P method comes from the element-local L2- projections, as well as the stabilization mechanism. However, we think that the latter is the Achilles’ heel of this method, since the numerical treatments involved in and are a little complex. In the original EL2P method, the authors introduced additional auxiliary shape functions, precisely the bubble functions associated with each face and/or each element to the finite element subspace . Thus we have to take more efforts to define those complex bubbles and take more time cost to derive the element stiff matrices in the finite element subroutines.
In this paper, we would like to present a new and simple treatment to achieve enough stability. This new development comes from the techniques on the element-local L2-projections, and the stabilization mechanism is established completely in the simple framework of least-squares method. Based on these ideas, we would like to seek the approximation solution , such that for any test function , there holds the variational formwhere a new element-local L2-projection is introduced here, in additional to two element-local L2-projections, and , same as those in the original EL2P method. The detailed implementation and discussion will be given in Section 2. For convenience, in this paper we would like to refer to the above method as the new EL2P method.
The stability mechanism based on the least-square method is the highlights of this new EL2P method. On comparison with the original method, the stability mechanism now is reduced to a compact form, which depends only on the element-local L2-projection and is independent of the given data f and g. As a result, the new method is simpler and easier to be implemented, and much time cost can be saved. Also, this simple implementation in the new EL2P method can be applied directly and easily to the numerical simulation on the Maxwell’s equations and the Maxwell’s eigenvalue problems. Further, this simple treatment in the new EL2P method still provides excellent performances for both the H1-solution and the non-H1-solution. In many numerical tests, the numerical result is almost same as or better than that given by the original EL2P method. For example, when this new method is applied to some benchmarks for the Maxwell’s equations and the Maxwell’s eigenvalues problem, the relative errors seem more superior to that given in [12]. Finally, it is worthy to mention that this new method also has the coercivity property and the optimal error estimate as the original EL2P method, in numerical experiments, although in this paper we only prove this issue for a special case.
The remaining part of this paper is organized as follows. Section 2 is the main body of this paper, where the new EL2P method is established for the electrostatics problem. Two theoretical conclusions about the coercivity property and the optimal error estimate for the non-H1 solution are then presented in a special case. The detailed proofs are derived in the appendix. Finally we present some numerical tests to verify the optimal convergence order and show the nice saving in time cost. As an extension and application of this new method, we would like in Section 3 to apply this new and simple implementation to resolve the Maxwell’s equations and its eigenvalue problems. The remark conclusion and future work are given in Section 4.
Section snippets
New EL2P method for electrostatic problem
We would like to present in this section the new EL2P method for electrostatic problem (1) in two dimension. The considered boundary condition is , where is the outward unit normal vector on the boundary . The necessary condition for the existence of solution is that the average of f in is equal to zero.
Applications
In this section we will apply the simple implementation used in the new EL2P method to resolve the Maxwell’s equations and the Maxwell’s eigenvalue problem. By some numerical tests on the famous benchmark problems, we will show this new method also works effectively and robustly.
Conclusion and future work
In this paper, we present a new and simple treatment to facilitate the original EL2P method for the electrostatic problems in two dimension, where the solution has a low regularity outside . This strategy can also be applied to the Maxwell’s equations and its eigenvalues computation. The numerical experiments are effective and robust, which show that this method has the optimal convergence order and better performance than the original one. The numerical results are still good even when
Acknowledgments
This work was supported in part by the National Science Foundation of CHINA under the Grants 11071132, 11171168 and 11271187, and Research Fund for the Doctoral Program of Higher Education of China under Grant 20100031110002.
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