The lumped mass finite element method is a kind of modified finite element method. It has the same convergence and error estimation as the traditional finite element method, but it has a smaller amount of calculation. Therefore, the lumped mass finite element method is favored by scholars at home and abroad. It is also one of the hot topics being studied  - . It is known that equations of this type arise in many areas of mathematical physics and fluid mechanics. It has been studied extensively by Benjamin and others, as a model for unidirectional, long, dispersive waves. It has been widely used in linear optics, iso-particle physics, etc. The numerical solution of problem has been studied in    , among them, the standard Galerkin method, the finite difference method and the general method are applied to this equation. Feng Minfu et al. proposed a Crank-Nicolson difference method to discretize the equation in . Khaled Omrani made a detailed analysis of the standard Galerkin method of this equation in : the space is discretized by the standard Galerkin, and the time discretization is in the Crank-Nicolson format, the convergence of the method is proved.
Tan Yanmei et al. applied the mixed finite element method to this equation in , established semi-discretization and full discretization finite element format, the existence and uniqueness of the finite element solution is proved, and an error analysis is given. However, the research on the lumped mass finite element method of nonlinear BBM equations has not been reported.
The main purpose of this article is to study the lumped mass finite element method for the BBM equation, and approximation using non-conforming triangle elements. Through the particularity of unit construction and interpolation techniques, the optimal error estimate is obtained without the need for traditional Reiz projection. In this paper, C denotes generic positive constant independent of step sizes and not necessarily the same at each occurrence.
2. Lumped Mass Nonconforming Fully Discrete Crank-Nicolson Scheme of BBM Equation
We will consider the following nonlinear BBM equation:
where is a bounded domain with smooth boundary , , and , .
For a nonnegative integer m, let denote the usual Sobolev space of real-valued functions defined on .
The norm of this space is the usual Sobolev -norm and it will be denoted by ,
where, , and are two non-negative integers , specially, let , denoted as , denote the inner product in .
We introduce the weak formulation of (1). Let by and use the Green formula.
Proposition 1. Let u be the solution of (2). Then, the following conservation of energy holds.
Proof. Setting in (2), we get
where . Since , on and , then
Form (4) and (5). It follows that,
Integrating (6) with respect to t, to complete the rest of the proof.
Let be a polygon on plane with boundaries parallel to the axes. be an triangle subdivision of . , which does not need to
satisfy the regularity assumption or quasi-uniform assumption . For a give , the three vertices of K are . Let be the reference element on plane, the three vertices of are .
For all , we define the finite element on as follows 
where , , represent the edge of .
It can be easily checked that interpolation defined above is well-posed and the interpolation function can be expressed as
Define the affine mapping by .
For all , then we define the associated finite element space and bilinear form as
where stands for the jump of v across the edge l, if l is an internal edge and it is equal to v itself if .
For the associated finite element interpolation is
For , we define . It is easy to see that is a norm over .
Now we consider the following numerical integration format
where is the three vertices of K.
According to the Bramble-Hilbert lemma  and (10) is exactly true for the linear polynomial, we get
From (11), we know and are equivalent in space , then there exist two constants and , independent of h and k such that
For any given positive integer N, let denote the size of the time discretization and , , . The linear function determined by the values of two nodes and is an approximate solution of , then lumped mass nonconforming fully discrete Crank-Nicolson scheme of (2) lets , to find , such that
where is an appropriate approximation to , , , , , .
According to the theoretical knowledge of the numerical solution of partial differential equations that problem (14) has a unique solution .
3. Error Estimates
For simplicity, let , , , ,
, , , .
Lemma 3.1. There exists a constant such that for all ,
Proof. , u and v are linear functions on the unit K, from (11), we have
summing k and using Cauchy-Schwarz inequality, we get
Lemma 3.2.  Suppose , then, under the anisotropic meshes, there holds
where is the outward unit vector of , let
Lemma 3.3. For all , the solution of (2), then
Proof. From (2), for all ,
Then using Green’s formula we have
from Lemma 3.2, for all , then
from (17), integral on both sides for , such that
let , we have
from (16), for all , then .
According to the one-dimensional linear interpolation theory and the Cauchy-Schwarz inequality, we get
which completes the proof.
Lemma 3.4. There exists a constant r, , for all L, and (L is a positive integer), then there holds
Proof. Subtracting (15) from the first formula of (14), for all , we have
according to the definition of and , from (19), we obtain
by the characteristics of the unit of C-R, then
further, using the second formula of (14), we get
let , substitute the above formula in (19), then
Now we shall respectively estimate the terms at the right end of the Equation (20), from Lemma 3.3 and Young inequality, we get
Above the second item, we obtain
From , the third item is estimated as follows
Substitute the above estimation results in (20), furthermore, from
For , so that
multiplied with both sides, then summing up from to , we get
Finally, use (23) and (24) to complete the rest of the proof.
Theorem 3.1. let and be the solutions of (2), suppose is sufficiently smooth, then there holds
Proof. Using the definition of and the triangle inequality, we get
then using Lemma 3.4 and the interpolation theorem, the proof is completed.
Key scientific research projects of colleges and universities in Henan Province (NO: 19A110031).
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