An Arbitrary-Order Discontinuous Galerkin Method with One Unknown Per Element

Abstract

We propose an arbitrary-order discontinuous Galerkin method for secondorder elliptic problem on general polygonal mesh with only one degree of freedom per element. This is achieved by locally solving a discrete least-squares over a neighboring element patch. Under a geometrical condition on the element patch, we prove an optimal a priori error estimates for the energy norm and for the L 2 norm. The accuracy and the efficiency of the method up to order six on several polygonal meshes are illustrated by a set of benchmark problems.

A least squares method for linear elasticity using a patch reconstructed space

“…Under some mild and practical conditions on element patch S(K), Λ m could be bounded uniformly which plays a vital role in the convergence estimate. We refer to [21,20] for these conditions and more detailed discussion about the uniform upper bound. One of the conditions we shall note is that the number #S(K) shall be far greater than dim(P m ).…”

“…In this paper, a new discontinuous least squares finite element method is proposed based on the stress-displacement formulation. The novel point is the new approximation space which is obtained by patch reconstruction with one unknown per element [20,21,19]. The new space could be regarded as a subspace of the common space used in discontinuous Galerkin finite element method.…”

A Sequential Least Squares Method for Poisson Equation Using a Patch Reconstructed Space

“…In this section, we follow the idea in [20,19] to define an approximation space using a patch reconstruction operator. Purposely, the reconstruction operator we propose here will use the irrotational basis, thus the approximation space obtained is piecewise rotation free.…”

“…We note that the existence of the solution to (11) is obvious but the uniqueness of the solution depends on the position of the sampling nodes in I K , here we follow [20] to state the following assumption:…”

“…In the first step, we approximate the flux still using a discontinuous approximation space. This space is the piecewise irrotational polynomial space, which is a generalization of the reconstructed space proposed in [21,20]. The new space is obtained by solving a local least squares problem based on the irrotational polynomial bases and only one unknown locates inside each element.…”