We consider the biharmonic equation of the form
where is a bounded polytopal domain in .
However, it is known that -conforming methods require -continuous piecewise polynomials on a simplicial meshes, which imposes difficulty in practical computation. Due to the complexity in the construction of -continuous elements, -conforming finite element methods are rarely used in practice for solving the biharmonic equation.
An approach of avoiding construction of -conforming elements is to use discontinuous approximations. Due to the flexibility of discontinuous Galerkin (DG) finite element methods in element constructions and in mesh generations, many finite element methods have been developed using totally discontinuous polynomials. Here we are only interested in interior penalty discontinuous Galerkin (IPDG) methods since the proposed the method shares the same finite element spaces with IPDG method. For the biharmonic equation, interior penalty discontinuous Galerkin finite element methods have been studied in [1, 2, 3, 4, 5, 8]. One obvious disadvantage of discontinuous finite element methods is their rather complicated formulations which are often necessary to guarantee well posedness and convergence of the methods. For example, the symmetric IPDG method for the biharmonic equation with homogenous boundary conditions [1, 2] has the following formulation:
where and are two parameters that need to be tuned.
The purpose of this work is to introduce a conforming DG finite element method for the biharmonic equation which has the following ultra simple formulation without any stabilizing/penalty terms and other mixed terms of lower dimension integrations in (6):
where is called weak Laplacian, an approximation of . The formulation (7) can be viewed as a counterpart of (5) for discontinuous approximations. The conforming DG method was first introduced in [12, 13] for second order elliptic equations, which, by name, means the method using the finite element spaces of DG methods and the simple formulations of conforming methods. This new finite element method shares the same finite element space with the IPDG methods but having much simpler formulation. This simple formulation can be obtained by defining weak Laplacian appropriately. The idea here is to raise the degree of polynomials used to compute weak Laplacian . Using higher degree polynomials in computation of weak Laplacian will not change the size, neither the global sparsity of the stiffness matrix. Optimal order error estimates in a discrete for and in norm for are established for the corresponding finite element solutions. Numerical results are provided to confirm the theories.
2 A Conforming DG Finite Element Method
Let be a partition of the domain consisting of polygons in two dimension or polyhedra in three dimension satisfying a set of conditions defined in  and additional conditions specified in . Denote by the set of all edges or flat faces in , and let be the set of all interior edges or flat faces.
For simplicity, we adopt the following notations,
Let consist all the polynomials degree less or equal to defined on .
We define a finite element space for as follows
Let and be two polygons/polyhedrons sharing if . Let and
be scalar and vector valued functions, the jumpsand are defined as
and the averages and are defined as
If is on , then
Weak Galerkin Algorithm 1
Next we will discuss how to compute the weak Laplacian and in (12). The concept of weak derivative was first introduced in [10, 9] for weak functions in weak Galerkin methods and was modified in [7, 11]. A weak Laplacian operator, denoted by , is defined as the unique polynomial for that satisfies the following equation
Let , then on any ,
where is a locally defined projections onto on each element . It is not hard to see that for any we have
We complete the proof.
3 Well Posedness
First we define a semi-norm as
Then we introduce a discrete norm as follows:
The following lemma indicates that the two norms and are equivalent.
First we need the following trace inequality. For any function , the trace inequality holds true (see  for details):
There exist two positive constants and such that for any , we have
For any , it follows from the definition of weak Laplacian (13) and integration by parts that
By letting in (20) we arrive at
It is easy to see that the following equations hold true for on with ,
Next we will prove
It follows from (20) that for any ,
By Lemma 3.1 in , there exist a such that for ,
Letting in (23) yields
Taking the summation of the above equation over and using (21), one has
Similarly, by Lemma 3.2 in , we can have
Finally, by letting in (23) we arrive at
We complete the proof.
The finite element scheme (12) has a unique solution.
4 An Error Equation
Let . Next we derive an error equation that satisfies.
For any , we have
Testing (1) by and using the fact that and and integration by parts, we arrive at
Combining the above two equations gives
which implies that
The error equation follows from subtracting (12) from the above equation,
We have proved the lemma.
5 An Error Estimate in
We start this section by defining some approximation operator. Let be the element-wise defined projection onto on each element .
Let and . There exists a constant such that the following estimates hold true:
Here is the usual Kronecker’s delta with value when and value otherwise.
Let , and . There exists a constant such that
We complete the proof.
Let , then
Taking the summation over , we have proved the lemma.
Let be the finite element solution arising from (12). Assume that the exact solution . Then, there exists a constant such that
Let . Then it is straightforward to obtain
which completes the proof.
6 Error Estimates in Norm
In this section, we will obtain an error bound for the finite element solution in norm.
The dual problem considered has the following form,
Assume that the regularity holds,
Testing (40) by and using the fact that and and integration by parts, we arrive at