Abstract Finite elements of higher continuity, say conforming in H 2 {H^{2}} instead of H 1 {H^{1}} , require a mapping from reference cells to mesh cells which is continuously differentiable across cell interfaces. In this article, we propose an algorithm to obtain such mappings given a topologically regular mesh in the standard format of vertex coordinates and a description of the boundary. A variant of the algorithm with orthogonal edges in each vertex is proposed. We introduce necessary modifications in the case of adaptive mesh refinement with nonconforming edges. Furthermore, we discuss efficient storage of the necessary data.

Abstract We introduce a new geometric multigrid algorithm to solve elliptic interface problems. First we discretize the problems by the usual P 1 {P_{1}} -conforming finite element methods on a semi-uniform grid which is obtained by refining a uniform grid. To solve the algebraic system, we adopt subspace correction methods for which we use uniform grids as the auxiliary spaces. To enhance the efficiency of the algorithms, we define a new transfer operator between a uniform grid and a semi-uniform grid so that the transferred functions satisfy the flux continuity along the interface. In the auxiliary space, the system is solved by the usual multigrid algorithm with a similarly modified prolongation operator. We show 𝒲 {\mathcal{W}} -cycle convergence for the proposed multigrid algorithm. We demonstrate the performance of our multigrid algorithm for problems having various ratios of parameters. We observe that the computational complexity of our algorithms are robust for all problems we tested.

Abstract A generalized thermistor model is discretized thanks to a fully implicit vertex-centered finite volume scheme on simplicial meshes. An assumption on the stiffness coefficients is mandatory to prove a discrete maximum principle on the electric potential. This property is fundamental to handle the stability issues related to the Joule heating term. Then the convergence to a weak solution is established. Finally, numerical results are presented to show the efficiency of the methodology and to illustrate the behavior of the temperature together with the electric potential within the medium.

Abstract A conforming triangular mixed element recently proposed by Hu and Zhang for linear elasticity is extended by rearranging the global degrees of freedom. More precisely, adaptive meshes 𝒯 1 , … , 𝒯 N {\mathcal{T}_{1},\ldots,\mathcal{T}_{N}} which are successively refined from an initial mesh 𝒯 0 {\mathcal{T}_{0}} through a newest vertex bisection strategy, admit a crucial hierarchical structure, namely, a newly added vertex 𝒙 e {\boldsymbol{x}_{e}} of the mesh 𝒯 ℓ {\mathcal{T}_{\ell}} is the midpoint of an edge e of the coarse mesh 𝒯 ℓ - 1 {\mathcal{T}_{\ell-1}} . Such a hierarchical structure is explored to partially relax the C 0 {C^{0}} vertex continuity of symmetric matrix-valued functions in the discrete stress space of the original element on 𝒯 ℓ {\mathcal{T}_{\ell}} and results in an extended discrete stress space: for such an internal vertex 𝒙 e {\boldsymbol{x}_{e}} located at the coarse edge e with the unit tangential vector t e {t_{e}} and the unit normal vector n e = t e ⊥ {n_{e}=t_{e}^{\perp}} , the pure tangential component basis function φ 𝒙 e ⁢ ( 𝒙 ) ⁢ t e ⁢ t e T {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})t_{e}t_{e}^{T}} of the original discrete stress space associated to vertex 𝒙 e {\boldsymbol{x}_{e}} is split into two basis functions φ 𝒙 e + ⁢ ( 𝒙 ) ⁢ t e ⁢ t e T {\varphi_{\boldsymbol{x}_{e}}^{+}(\boldsymbol{x})t_{e}t_{e}^{T}} and φ 𝒙 e - ⁢ ( 𝒙 ) ⁢ t e ⁢ t e T {\varphi_{\boldsymbol{x}_{e}}^{-}(\boldsymbol{x})t_{e}t_{e}^{T}} along edge e, where φ 𝒙 e ⁢ ( 𝒙 ) {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})} is the nodal basis function of the scalar-valued Lagrange element of order k (k is equal to the polynomial degree of the discrete stress) on 𝒯 ℓ {\mathcal{T}_{\ell}} with φ 𝒙 e + ⁢ ( 𝒙 ) {\varphi_{\boldsymbol{x}_{e}}^{+}(\boldsymbol{x})} and φ 𝒙 e - ⁢ ( 𝒙 ) {\varphi_{\boldsymbol{x}_{e}}^{-}(\boldsymbol{x})} denoted its two restrictions on two sides of e, respectively. Since the remaining two basis functions φ 𝒙 e ⁢ ( 𝒙 ) ⁢ n e ⁢ n e T {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})n_{e}n_{e}^{T}} , φ 𝒙 e ⁢ ( 𝒙 ) ⁢ ( n e ⁢ t e T + t e ⁢ n e T ) {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})(n_{e}t_{e}^{T}+t_{e}n_{e}^{T})} are the same as those associated to 𝒙 e {\boldsymbol{x}_{e}} of the original discrete stress space, the number of the global basis functions associated to 𝒙 e {\boldsymbol{x}_{e}} of the extended discrete stress space becomes four rather than three (for the original discrete stress space). As a result, though the extended discrete stress space on 𝒯 ℓ {\mathcal{T}_{\ell}} is still a H ⁢ ( div ) {H(\operatorname{div})} subspace, the pure tangential component along the coarse edge e of discrete stresses in it is not necessarily continuous at such vertices like 𝒙 e {\boldsymbol{x}_{e}} . A feature of this extended discrete stress space is its nestedness in the sense that a space on a coarse mesh 𝒯 {\mathcal{T}} is a subspace of a space on any refinement 𝒯 ^ {\hat{\mathcal{T}}} of 𝒯 {\mathcal{T}} , which allows a proof of convergence of a standard adaptive algorithm. The idea is extended to impose a general traction boundary condition on the discrete level. Numerical experiments are provided to illustrate performance on both uniform and adaptive meshes.

Abstract In matrix-vector multiplication, matrix symmetry does not permit a straightforward reduction in computational cost. More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms. We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. In particular, it lowers the bilinear complexity of symmetrized contractions of symmetric tensors of order s + v {s+v} and v + t {v+t} by a factor of ( s + t + v ) ! s ! ⁢ t ! ⁢ v ! {\frac{(s+t+v)!}{s!t!v!}} to leading order. The algorithm computes a symmetric tensor of bilinear products, then subtracts unwanted parts of its partial sums. Special cases of this algorithm provide improvements to the bilinear complexity of the multiplication of a symmetric matrix and a vector, the symmetrized vector outer product, and the symmetrized product of symmetric matrices. While the algorithm requires more additions for each elementwise product, the total number of operations is in some cases less than classical algorithms, for tensors of any size. We provide a round-off error analysis of the algorithm and demonstrate that the error is not too large in practice. Finally, we provide an optimized implementation for one variant of the symmetry-preserving algorithm, which achieves speedups of up to 4.58 × \times for a particular tensor contraction, relative to a classical approach that casts the problem as a matrix-matrix multiplication.

The highly selective international mathematical journal Computational Methods in Applied Mathematics (CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs. CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics.

Abstract In this article, we study the numerical approximation of a variable coefficient fractional diffusion equation. Using a change of variable, the variable coefficient fractional diffusion equation is transformed into a constant coefficient fractional diffusion equation of the same order. The transformed equation retains the desirable stability property of being an elliptic equation. A spectral approximation scheme is proposed and analyzed for the transformed equation, with error estimates for the approximated solution derived. An approximation to the unknown of the variable coefficient fractional diffusion equation is then obtained by post-processing the computed approximation to the transformed equation. Error estimates are also presented for the approximation to the unknown of the variable coefficient equation with both smooth and non-smooth diffusivity coefficient and right-hand side. Numerical experiments are presented to test the performance of the proposed method.

Abstract We propose novel ensemble calculation methods for Navier–Stokes equations subject to various initial conditions, forcing terms and viscosity coefficients. We establish the stability of the schemes under a CFL condition involving velocity fluctuations. Similar to related works, the schemes require solution of a single system with multiple right-hand sides. Moreover, we extend the ensemble calculation method to problems with open boundary conditions, with provable energy stability.

The highly selective international mathematical journal Computational Methods in Applied Mathematics (CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs. CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics.

Abstract Richardson Extrapolation is a very general numerical procedure, which can be applied in the solution of many mathematical problems in an attempt to increase the accuracy of the results. It is assumed that this approach is used to handle non-linear systems of ordinary differential equations (ODEs) which arise often in the mathematical description of scientific and engineering models either directly or after the discretization of the spatial derivatives of partial differential equations (PDEs). The major topic is the analysis ofeightadvanced implementations of the Richardson Extrapolation. Two important properties are analyzed: (a) the possibility to achievemore accurate resultsand (b) the possibility to improvethe stability propertiesof eight advanced versions of the Richardson Extrapolation. A two-parameter family of test-examples was constructed and used to check both the accuracy and the absolute stability of the different versions of the Richardson Extrapolation when these versions are applied together with several Explicit Runge–Kutta Methods (ERKMs).