New estimates of mixed finite element method for fourth‐order wave equation

Abstract

Communicated by S. WiseThe conforming bilinear mixed finite element method is established for a fourth-order wave equation. By applying the interpolation and projection simultaneously, the superclose and superconvergence results of order O.h 2 / for the original variable u and intermediate variable p D a 1 .X/u in H 1 -norm are achieved for the semi-discrete and fully discrete schemes, respectively (where a 1 .X/ is the coefficient appeared in the equation). The distinct advantage of this method is that it can reduce the smoothness of solutions u, u t , and p compared with the existing literature for getting optimal estimates alone. At last, some numerical results are carried out to validate the theoretical analysis. 4448D. SHI ET AL. projection. Secondly, through establishing the relationship between the Ritz projection and interpolation of the bilinear element, the superclose estimate of the interpolation for u in H 1 -norm is obtained. Thirdly, the superclose property of the interpolation for p in H 1norm is obtained under the condition a 1 .X/ D 1 c.X/ . Then the global superconvergence results for the given two variables are derived with the help of the interpolation post-processing technique. At last, some numerical results are provided to illustrate the validity of the theoretical analysis and effectiveness of the proposed method.It should be pointed out that the superclose result of order O.h 2 / for u in H 1 -norm and convergence result for p in L 2 -norm are obtained by use of the interpolation operator alone in [6], the solutions u, u t , and p are required belonging to H 4 . /. However, in this paper, by applying the interpolation and projection simultaneously, we prove the superclose results of order O.h 2 / for both u and p in H 1 -norm under weaker assumptions u, u t , p 2 H 3 . /. On the other hand, if the projection is utilized alone, we cannot derive the global superconvergence results for how to construct a suitable post-processing operator for the projection still remains open.The outline of this paper is organized as follows: In Section 2, the MFE spaces and a lemma are introduced. In Sections 3 and 4, the superclose and superconvergence results are proved for the semi-discrete and fully discrete schemes, respectively. In Section 5, some remarks are given. In the last section, two numerical examples are presented.Throughout this article, C denotes a positive constant that may take different values at different places but remains independent of the mesh parameter h.C .a 2 .X/rÂ, r /. Remark 4From the proofs of Theorems 3.3 and 4.3, we can find that the constraints on a 2 .X/ and a 3 .X/ are stronger than that in the paper [6], and we derive the superclose property of p under the condition c.X/ D b.X/; how to get rid of this restriction and extend to the general situation is one of topics in our further work. Numerical resultsExample 6.1 Consider the following constant coefficient equation (a 1 .X/ D a 2 .X/ D a 3 .X/ D c.X/ D 1) u tt C 2 u u C u D f .X, t/, in .0, T/, @u @n D @.u/ @n D 0, on @...

Uniform superconvergence analysis of Ciarlet‐Raviart scheme for Bi‐wave singular perturbation problem

“…respectively. According to the estimates of Shi et al, 22 for v h ∈ V h , u ∈ H 4 (Ω), we can prove that…”

“…As we know, the approximate spaces of the standard conforming FEMs for solving the fourth‐order PDEs are usually required to be included in H 2 (Ω) . For getting rid of this complexity, the mixed FEM has been developed in previous studies . But, up to now, there are no studies having much to do with uniform convergence analysis of mixed FEM for the Bi‐wave SPP .…”

“…16,17 For getting rid of this complexity, the mixed FEM has been developed in previous studies. [18][19][20][21][22][23] But, up to now, there are no studies having much to do with uniform convergence analysis of mixed FEM for the Bi-wave SPP 1.…”

High Accuracy Analysis of Nonconforming Mixed Finite Element Method for the Nonlinear Sivashinsky Equation

“…e nonconforming MFEM brings down the smoothness requirement on FE solution compared to the conforming case. Readers with more interests may refer [11][12][13][14][15] and the references listed. For problem (1), Omrani [1] developed the convergence analysis of the corresponding variables in the semidiscrete and fullydiscrete schemes by using conforming MFEM; however, situation involving nonconforming MFEM was not available till now.…”

“…Meanwhile, superconvergence is critical in practical engineering numerical calculation and has always been a research hotspot. To find out more applications, readers may refer [12,[15][16][17][18][19][20][21][22][23]. As far as our knowledge is concerned, research on superconvergence for Sivashinsky equation is yet to be found.…”