$P_1$--nonconforming polyhedral **finite** **elements** in high dimensions

**finite**

**element**methods for the approximation of elliptic problems in high dimensions. The $P_1$--nonconforming polyhedral

**finite**

**element**is introduced for any high dimension. Our

**finite**

**element**is simple and cheap as it is based on the triangulation of domains into polytopes, which are combinatorially equivalent to $d$--dimensional cube, rather than the triangulation of domains into simplices. Our nonconforming

**element**is nonparametric, and on each polytope it contains only linear polynomials, but it is sufficient to give optimal order convergence for second--order elliptic problems.

10/10 relevant

arXiv

A Moving Mesh Method for Modelling Defects in Nematic Liquid Crystals

**finite**

**element**based adaptive moving mesh approach, where an unstructured triangular mesh is adapted towards high activity regions, including those around defects. Spatial convergence studies are presented using a stationary defect as a model test case, and the adaptive method is shown to be optimally convergent using quadratic triangular

**finite**

**elements**. The full effectiveness of the method is then demonstrated using a challenging two-dimensional dynamic Pi-cell problem involving the creation, movement, and annihilation of defects.

7/10 relevant

arXiv

A Conservative **Finite** **Element** ALE Scheme for Mass-Conserving
Reaction-Diffusion Equations on Evolving Two-Dimensional Domains

**finite**

**element**solution is established independently of the ALE velocity field and the time step size. Expand abstract.

**finite**

**element**method for the approximate solution of systems of bulk-surface reaction-diffusion equations on an evolving two-dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a moving mesh partial differential equation (MMPDE) approach. Global conservation of the fully discrete

**finite**

**element**solution is established independently of the ALE velocity field and the time step size. The developed method is applied to model problems with known analytical solutions; these experiments indicate that the method is second-order accurate and globally conservative. The method is further applied to a model of a single cell migrating in the presence of an external chemotactic signal.

4/10 relevant

arXiv

A Compatible **Finite** **Element** Discretisation for the Moist Compressible
Euler Equations

**finite**

**element**framework identified in Cotter and Shipton (2012). Expand abstract.

**finite**

**element**framework identified in Cotter and Shipton (2012). The discretisation strategy is described and details of the parametrisations of moist processes are presented. A procedure for establishing hydrostatic balance for moist atmospheres is introduced, and the model's performance is demonstrated through several test cases, two of which are new.

10/10 relevant

arXiv

A high accuracy nonconforming **finite** **element** scheme for Helmholtz
transmission eigenvalue problem

**finite**

**element**scheme $B_{h0}^3$ which does not correspond to a locally defined

**finite**

**element**with Ciarlet$'$s triple but admit a set of local basis functions. For the first time, we deduce and write out the expression of basis functions explicitly. Distinguished from the most nonconforming

**finite**

**element**methods, $(\delta\Delta_h\cdot,\Delta_h\cdot)$ with non-constant coefficient $\delta>0$ is coercive on the nonconforming $B_{h0}^3$ space which makes it robust for numerical discretization. For fourth order eigenvalue problem, the $B_{h0}^3$ scheme can provide $\mathcal{O}(h^2)$ approximation for the eigenspace in energy norm and $\mathcal{O}(h^4)$ approximation for the eigenvalues. We test the $B_{h0}^3$ scheme on the vary-coefficient bi-Laplace source and eigenvalue problem, further, transmission eigenvalue problem. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed scheme.

10/10 relevant

arXiv

Isogeometric analysis in option pricing

**finite**

**element**method is especially useful for practitioners dealing with derivatives where closed-form solution is not available. Expand abstract.

**finite**

**element**analysis directly into design described by non-uniform rational B-splines (NURBS). In this paper we show that price surfaces that occur in option pricing can be easily described by NURBS surfaces. For a class of stochastic volatility models, we develop a methodology for solving corresponding pricing partial integro-differential equations numerically by isogeometric analysis tools and show that a very small number of space discretization steps can be used to obtain sufficiently accurate results. Presented solution by

**finite**

**element**method is especially useful for practitioners dealing with derivatives where closed-form solution is not available.

5/10 relevant

arXiv

$H^1$-norm error estimate for a nonstandard **finite** **element** approximation
of second-order linear elliptic PDEs in non-divergence form

**finite**

**element**method for approximating $H^2$ strong solutions of second order linear elliptic PDEs in non-divergence form with continuous coefficients. To circumvent the difficulty of lacking an effective duality argument for this class of PDEs, a new analysis technique is introduced; the crux of it is to establish an $H^1$-norm stability estimate for the

**finite**

**element**approximation operator which mimics a similar estimate for the underlying PDE operator recently established by the authors and its proof is based on a freezing coefficient technique and a topological argument. Moreover, both the $H^1$-norm stability and error estimate also hold for the linear

**finite**

**element**method.

10/10 relevant

arXiv

An adaptive edge-based smoothed **finite** **element** method (ES-FEM) for
phase-field modeling of fractures at large deformations

**finite**

**element**method (FEM), which is characterized by higher accuracy, softer stiffness, and insensitive to mesh distortion. Expand abstract.

**finite**

**element**method (ES-FEM) for the first time. Therein the phase-field modeling of fractures has attracted widespread interest by virtue of its outstanding performance in dealing with complex cracks. The ES-FEM is an excellent member of the S-FEM family developed in combination with meshless ideas and

**finite**

**element**method (FEM), which is characterized by higher accuracy, softer stiffness, and insensitive to mesh distortion. Given that, the advantages of the phase-field method (PFM) and ES-FEM are fully combined by the approach proposed in this paper. With the costly computational overhead of PFM and ES-FEM in mind, a well-designed multi-level adaptive mesh strategy was developed, which considerably improved the computational efficiency. Furthermore, the detailed numerical implementation for the coupling of PFM and ES-FEM is outlined. Several representative numerical examples were recalculated based on the proposed method, and its effectiveness is verified by comparison with the results in experiments and literature. In particular, an experiment in which cracks deflected in rubber due to impinging on a weak interface was firstly reproduced.

10/10 relevant

arXiv

Galerkin **Finite** **Element** Method for Nonlinear Riemann-Liouville and
Caputo Fractional Equations

**finite**

**element**approach to proceed the numerical approximation of the weak formulations and prove a priori error estimations. Expand abstract.

**finite**

**element**approach to proceed the numerical approximation of the weak formulations and prove a priori error estimations. Finally, some numerical experiments are provided to explain the accuracy of the proposed method.

10/10 relevant

arXiv

Higher order Trace **Finite** **Element** Methods for the Surface Stokes
Equation

**finite**

**element**technique, known from the literature on trace finite element methods for scalar surface partial differential equations. Expand abstract.

**finite**

**element**methods for the discretization of surface Stokes equations is studied. These methods are based on an unfitted

**finite**

**element**approach in which standard Taylor-Hood spaces on an underlying bulk mesh are used. For treating the constraint that the velocity must be tangential to the surface a penalty method is applied. Higher order geometry approximation is obtained by using a parametric trace

**finite**

**element**technique, known from the literature on trace

**finite**

**element**methods for scalar surface partial differential equations. Based on theoretical analyses for related problems, specific choices for the parameters in the method are proposed. Results of a systematic numerical study are included in which different variants are compared and convergence properties are illustrated.

10/10 relevant

arXiv