MSTuD28
Weak Galerkin Method and Its Applications  Part III of III
For Part I, see MSMoD28
For Part II, see MSMoE28
Date: August 11
Time: 13:3015:30
Room: 109
(Note: Click title to show the abstract.)
Organizer:
Chen, Long (Univ. of California at Irvine)
Ye, Xiu (Univ. of Arkansas at Little Rock)
Zhang, Ran (Jilin Univ.)
Abstract: The Weak Galerkin method is an extension of the standard Galerkin finite element method where classical derivatives were substituted by weakly defined derivatives on functions with discontinuity. As such, the WG methods have the flexibility in handling complex geometry and low regularity solutions, the simplicity in analyzing realworld physical problems, and the symmetry in reformulating the original PDEs.
The aim of this minisymposium is to bring together specialists in order to ex change ideas regarding the development of WGFEMs and its industry and research applications. Since women is an underrepresented group in mathematics and engi neering, we pay a particular attention to attract female participants.
MSTuD281
13:3014:00
Equivalence of Weak Galerkin Methods and Virtual Element Methods for Elliptic Equations
Chen, Long (Univ. of California at Irvine)
Abstract: We propose a modification of the weak Galerkin methods and show its equivalence to a new version of virtual element methods. We also show the original weak Galerkin method is equivalent to the nonconforming virtual element method. As a consequence, ideas and techniques used for one method can be transferred to another. The key of the connection is the degree of freedoms.
MSTuD282
14:0014:30
A divergencefree weak Galerkin finite element
Zhang, Shangyou (Univ. of Delaware)
Abstract: A weak Galerkin finite element is designed so that the computed velocity is
divergencefree.
The significance of such a method is shown by solving a lowviscosity Stokes
problem.
The traditional finite elements, weak Galerkin finite elements and
discontinous Galerkin finite flements fail to produce
a meaningful solution in solving such a test problem.
CPTuD283
14:3014:50
Minisymposium's code "ycGz6P":
Modified weak Galerkin methods for convectiondiffusion problem
Gao, Fuzheng (Shandong Univ.)
Abstract: Minisymposium's code "ycGz6P":
In modern numerical simulation of problems in energy resources and environmental science, it is very important to develop efficient numerical methods for convection¨Cdiffusion problems. Based on modified weak gradient operator and weak divergence operator, we present a modified weak Galerkin finite element method ( MWGFEM) on arbitrary grids. Some techniques, such as calculus of variations, commutating operator and the theory of prior error estimates and techniques, are adopted. Optimal order error estimates for the corresponding MWGFEM approximations in both a discrete $H^1$ norm and the standard $L^2$ norm are derived to determine the errors in the approximate solution. Numerical results are presented to demonstrate the robustness, reliability, and accuracy of the MWGFEM
CPTuD284
14:5015:10
An adaptively weighted Galerkin finite element method for boundary value problems
Sun, Yifei (Courant Inst. of Mathematical Sci., New York Univ.)
Westphal, Chad (Wabash College)
Abstract: We introduce an adaptively weighted Galerkin approach for elliptic problems where diffusion is dominated by strong convection or reaction terms. In such problems, standard Galerkin approximations can have unacceptable oscillatory behavior near boundaries unless the computational mesh is sufficiently fine. Here we show how adaptively weighting the equations within the variational problem can increase accuracy and stability of solutions on underresolved meshes. Rather than relying on specialized finite elements or meshes, the idea here sets a flexible and robust framework where the metric of the variational formulation is adapted by an approximate solution. We give a general overview of the formulation and an algorithmic structure for choosing weight functions. Numerical examples are presented to illustrate the method.
CPTuD285
15:1015:30
hpVersion discontinuous Galerkin methods for partial differential
equation with nonnegative characteristic form on polygonal and
polyhedral meshes
Dong, Zhaonan (Univ. of Leicester)
Abstract: In this work, we consider hpversion discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of partial differential equation with nonnegative characteristic form on general computational meshes consisting of polygonal/polyhedral element. In particular, new hpa priori error bounds are derived in this work which improves the hpbounds in the work by [P.Houston, C.Schwab, E.S¨šli  Discontinuous hpFinite Element Methods for AdvectionDiffusionReaction Problems, SIAM Journal on Numerical Analysis 39(6):21332163, 2002]. The presented method employs elemental polynomial bases of total degree P defined on the physical space, without the need to map from a given reference or canonical frame. Numerical experiments highlighting the performance of the proposed method are presented. Moreover, the competitiveness of the DGFEM employing total degree P basis in comparison to the DGFEM employing Q basis on tensorproduct elements is studied numerically. This is the joint work with Andrea Cangiani, Emmanuil H. Georgoulis and Paul Houston.
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Footnote: Code: TypeDateTimeRoom No.
Type : IL=Invited Lecture, SL=Special Lectures, MS=Minisymposia, IM=Industrial Minisymposia, CP=Contributed Papers, PP=Posters
Date: Mo=Monday, Tu=Tuesday, We=Wednesday, Th=Thursday, Fr=Friday
Time : A=8:309:30, B=10:0011:00, C=11:1012:10, BC=10:0012:10, D=13:3015:30, E=16:0018:00, F=19:0020:00, G=12:1013:30, H=15:3016:00
Room No.: TBA
