Youngmok Jeon

A Hybrid Discontinuous Galerkin Method for Elliptic Problems

A new family of hybrid discontinuous Galerkin methods is studied for second-order elliptic equations. Our proposed method is a generalization of the cell boundary element (CBE) method [Y. Jeon and E.-J. Park, Appl. Numer. Math., 58 (2008), pp. 800-814], which allows high order polynomial approximations. Our method can be viewed as a hybridizable discontinuous Galerkin method [B. Cockburn, J. Gopalakrishnan, and R. Lazarov, SIAM J. Numer. Anal., 47 (2009), pp. 1319-1365] using a Bauman-Oden-type local solver. The method conserves the mass in each element and the average flux is continuous across the interelement boundary for even-degree polynomial approximations. Optimal order error estimates measured in the energy norm are proved. Numerical examples are presented to show the performance of the method.

Analysis of a cell boundary element method

Abstract The CBEM (cell boundary element method) was proposed as a numerical method for second-order elliptic problems by the first author in the earlier paper [10]. In this paper we prove a quasi-optimal order of convergence of the method, O(h1−ɛ) for ɛ>0 in H1-norm for the triangular mesh; also a stability result is obtained. We provide numerical examples and it is observed that the method conserves flux exactly when a certain condition on meshes is satisfied.

An indirect boundary integral equation method for the biharmonic equation

An integral equation method for the Dirichlet problem for the biharmonic equation is proposed. It leads to a

New locally conservative finite element methods on a rectangular mesh

2 \times 2

New boundary element formulas for the biharmonic equation

matrix integral equation system. By taking suitable norms on the spaces of density functions, the Fredholm operator theory can be used to prove the solvability. The kernels in this system are relatively complicated. Therefore, especially when a high-order polynomial approximation is used for a numerical purpose, it is costly to evaluate the integrals that appear in the numerical system. A discrete Galerkin method that has shown superb convergence is proposed here, as in [K. Atkinson, J. Integral Equations Appl., 1 (1988), pp. 343–363] and elsewhere. When the boundary functions are smooth, exponential convergence is observed.

Discrete qualocation methods for logarithmic-kernel integral equations on a piecewise smooth boundary

A new family of locally conservative, finite element methods for a rectangular mesh is introduced to solve second-order elliptic equations. Our approach is composed of generating PDE-adapted local basis and solving a global matrix system arising from a flux continuity equation. Quadratic and cubic elements are analyzed and optimal order error estimates measured in the energy norm are provided for elliptic equations. Next, this approach is exploited to approximate Stokes equations. Numerical results are presented for various examples including the lid driven-cavity problem.

A class of nonparametric DSSY nonconforming quadrilateral elements

We propose two new boundary integral equation formulas for the biharmonic equation with the Dirichlet boundary data that arises from plate bending problems in ℝ2. Two boundary conditions, u and ∂u/∂n, usually yield a 2 × 2 non-symmetric matrix system of integral equations. Our new formulas yield scalar integral equations that can be handled more efficiently for theoretical and numerical purposes. In this paper we supply complete ellipticity and solvability analyses of our new formulas. Numerical experiments for simple Galerkin methods are also provided.

New indirect scalar boundary integral equation formulas for the biharmonic equation

We consider a fully discrete qualocation method for Symm’s integral equation. The method is that of Sloan and Burn (1992), for which a complete analysis is available in the case of smooth curves. The convergence for smooth curves can be improved by a subtraction of singularity (Jeon and Kimn, 1996). In this paper we extend these results for smooth boundaries to polygonal boundaries. The analysis uses a mesh grading transformation method for Symm’s integral equation, as in Elschner and Graham (1995) and Elschner and Stephan (1996), to overcome the singular behavior of solutions at corners.

Hybrid Difference Methods for PDEs

A new class of nonparametric nonconforming quadrilateral finite elements is introduced which has the midpoint continuity and the mean value continuity at the interfaces of elements simul- taneously as the rectangular DSSY element (J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, ESAIM: M2AN 33 (1999) 747-770.) The parametric DSSY element for general quadrilaterals requires five de- grees of freedom to have an optimal order of convergence (Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Calcolo 37 (2000) 253-254.), while the new nonparametric DSSY elements require only four degrees of freedom. The design of new elements is based on the decomposition of a bilinear transform into a simple bilinear map followed by a suitable affine map. Numerical results are presented to compare the new elements with the parametric DSSY element. Mathematics Subject Classification. 65N30.

Hybrid Spectral Difference Methods for an Elliptic Equation

We derive scalar boundary integral equation formulas for both interior and exterior biharmonic equations with the Dirichlet boundary data. They are based on indirect boundary integral equation formulas, so-called the Chakrabarty and Almansi formulas. The scalar formulas are derived through an unconventional variational approach. The unique solvability results of the formulas are also obtained.