A Mixed Method for the Biharmonic Problem Based On a System of First-Order Equations

Abstract

Abstract. We introduce a new mixed method for the biharmonic problem. The method is based on a formulation where the biharmonic problem is re-written as a system of four first-order equations. A hybrid form of the method is introduced which allows to reduce the globally coupled degrees of freedom to only those associated with Lagrange multipliers which approximate the solution and its derivative at the faces of the triangulation. For k ≥ 1 a projection of the primal variable error superconverges with order k + 3 while the error itself converges with order k + 1 only. This fact is exploited by using local postprocessing techniques that produce new approximations to the primal variable converging with order k + 3. We provide numerical experiments that validate our theoretical results.

A Superconvergent $$C^0$$ C 0 Discontinuous Galerkin Method for Kirchhoff Plates: Error Estimates, Hybridization and Postprocessing

“…In Ref. [5], in light of the first order system related to the Hermann-Miyoshi formulation, a new mixed method for biharmonic problems was introduced with optimal error estimates O(h k+1 ) for all variables, where polynomials of degree k were used to approximate ∇u and u, and kth Raviart-Thomas element spaces were used to approximate σ and ∇ · σ , respectively. Moreover, by means of piecewise postprocessing technique, a new superconvergent approximation of u was obtained with the error O(h k+3 ) in L 2 norm.…”

“…Moreover, by means of piecewise postprocessing technique, a new superconvergent approximation of u was obtained with the error O(h k+3 ) in L 2 norm. It should be pointed out that the finite element space used to approximate σ in [5] is nonsymmetric though σ itself is symmetric. The other point is that both the HHJ method and the mixed method in [5] have made use of the technique of hybridization to reduce the globally coupled degrees of freedom, only leave behind unknowns related to the Lagrange multiplier function and the approximate deflection function u h , which decreases the size of the resulting linear system greatly.…”

“…It should be pointed out that the finite element space used to approximate σ in [5] is nonsymmetric though σ itself is symmetric. The other point is that both the HHJ method and the mixed method in [5] have made use of the technique of hybridization to reduce the globally coupled degrees of freedom, only leave behind unknowns related to the Lagrange multiplier function and the approximate deflection function u h , which decreases the size of the resulting linear system greatly. Now, let us discuss our CDG method in [32] which is also built upon the Hermann-Miyoshi formulation, with the variables σ and u approximated by piecewise polynomials of degrees l and k, respectively.…”

A super penalty C⁰ discontinuous Galerkin method for Kirchhoff plates

“…For this reason, Wells and Dung introduced in [3] a C 0 DG (CDG) method whose stability condition can be precisely quantified. In [4], we devised a class of stable CDG methods for Kirchhoff plate bending problems, with the numerical traces determined by a discrete stability identity and dependent on two nonnegative parameters C 11 and C 22 . If C 11 > 0 and C 22 = 0, we then obtained in the previous paper the local CDG (LCDG) method which does not contain any to-be-determined parameters and developed error analysis for the method based on its primal formulation.…”

“…Due to the difficulty in constructing H (div, , S)conforming elements, Hellan-Herrmann-Johnson (HHJ) method raised in [19][20][21] is a preferred mixed finite element for Kirchhoff plate bending problems in history, whose convergence rates for the primal variable u and the dual variable σ are both optimal. To obtain optimal convergence rate O(h k+1 ) of the approximation to ∇ 2 u, Behrens and Guzmán introduced a new mixed method which is also based on a system of first-order equations in [22]. And a hybrid technique is used for this mixed method to reduce the globally coupled degrees of freedom to only those associated with Lagrange multipliers, which is very efficient in implementation.…”