Nonconforming finite element Stokes complexes in three dimensions

Two nonconforming finite element Stokes complexes starting from the conforming Lagrange element and ending with the nonconforming P_1-P_0 element for the Stokes equation in three dimensions are studied.Commutative diagrams are also shown by combining nonconforming finite element Stokes complexes and interpolation operators.The lower order H(gradcurl)-nonconforming finite element only has 14 degrees of freedom,whose basis functions are explicitly given in terms of the barycentric coordinates.The H(gradcurl)-nonconforming elements are applied to solve the quad-curl problem,and the optimal convergence is derived.By the nonconforming finite element Stokes complexes,the mixed finite element methods of the quad-curl problem are decoupled into two mixed methods of the Maxwell equation and the nonconforming P_1-P_0 element method for the Stokes equation,based on which a fast solver is discussed.Numerical results are provided to verify the theoretical convergence rates.

BUCKLING ANALYSIS UNDER COMBINED LOADING OF THIN-WALLED PLATE ASSEMBLIES USING BUBBLE FUNCTIONS

Bubble functions are finite element modes that are zero on the boundary of the element but nonzero at the other point. The present paper adds bubble functions to the ordinary Complex Finite Strip Method(CFSM) to calculate the elastic local buckling stress of plates and plate assemblies. The results indicate that the use of bubble functions greatly improves the convergence of the Finite Strip Method(FSM) in terms of strip subdivision, and leads to much smaller storage required for the structure stiffness and stability matrices. Numerical examples are given, including plates and plate structures subjected to a combination of longitudinal and transverse compression, bending and shear. This study illustrates the power of bubble functions in solving stability problems of plates and plate structures.

A new method to calculate near-field radiation excited by heterogeneous fault rupture

This paper derives from the representation theory the formula for calculating the radiation excited by heterogeneous fault rupture based on box-like discretization scheme. Preliminary validation indicates that our algorithm has very high computation precision and efficiency; therefore, it is a very practical tool to investigate strong ground motion problems. Additionally, the equations given in this study can also be used to invert the fault rupture process.

Nonconforming Finite Elements for the -curlΔcurl and Brinkman Problems on Cubical Meshes

We propose two families of nonconforming elements on cubical meshes:one for the -curlΔcurl problem and the other for the Brinkman problem.The element for the -curlΔcurl problem is the first nonconforming element on cubical meshes.The element for the Brinkman problem can yield a uniformly stable finite element method with respect to the viscosity coefficient ν.The lowest-order elements for the -curlΔcurl and the Brinkman problems have 48 and 30 DOFs on each cube,respectively.The two families of elements are subspaces of H(curl;Ω)and H(div;Ω),and they,as nonconforming approximation to H(gradcurl;Ω)and[H^(1)(Ω)]^(3),can form a discrete Stokes complex together with the serendipity finite element space and the piecewise polynomial space.

Categorical resolutions of a class of derived categories

We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D^b(A)and the subcategory K^b(P) of perfect complexes in D^b(A), by giving two classes of abelian categories A with enough projective objects such that D_(hf)~b(A) = K^b(P), and finding an example such that D_(hf)~b(A)≠K^b(P). We realize the bounded derived category D^b(A) as a Verdier quotient of the relative derived category D_C^b(A), where C is an arbitrary resolving contravariantly finite subcategory of A. Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension.We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dimT <∞ such that ~⊥T is finite, then D^b(modA) admits a categorical resolution; and that for a CM(Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant.

Thompson's Group and the Linear Group GL_∞(Z)

The authors study the finite decomposition complexity of metric spaces of H, equipped with different metrics, where H is a subgroup of the linear group GL∞（E）. It is proved that there is an injective Lipschitz map φ： （F, ds） --＊ （H,d）, where F is the Thompson＇s group, ds the word-metric of F with respect to the finite generating set S and d a metric of H. But it is not a proper map. Meanwhile, it is proved that φ（F, ds） → （H, dl） is not a Lipschitz map, where dl is another metric of H.