Volume 21, Issue 1

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.

In this article, we extend the adaptive cross approximation (ACA) method known for the efficient approximation of discretisations of integral operators to a block-adaptive version. While ACA is usually employed to assemble hierarchical matrix approximations having the same prescribed accuracy on all blocks of the partition, for the solution of linear systems, it may be more efficient to adapt the accuracy of each block to the actual error of the solution as some blocks may be more important for the solution error than others. To this end, error estimation techniques known from adaptive mesh refinement are applied to automatically improve the blockwise matrix approximation. This allows to interlace the assembling of the coefficient matrix with the iterative solution.

Local discontinuous Galerkin method is considered for time-dependent singularly perturbed semilinear problems with boundary layer. The method is equipped with a general numerical flux including two kinds of independent parameters. By virtue of the weighted estimates and suitably designed global projections, we establish optimal ( k + 1 ) {(k+1)} -th error estimate in a local region at a distance of 𝒪 ⁢ ( h ⁢ log ⁡ ( 1 h ) ) {\mathcal{O}(h\log(\frac{1}{h}))} from domain boundary. Here k is the degree of piecewise polynomials in the discontinuous finite element space and h is the maximum mesh size. Both semi-discrete LDG method and fully discrete LDG method with a third-order explicit Runge–Kutta time-marching are considered. Numerical experiments support our theoretical results.

We represent the solution u ⁢ ( t ) {u(t)} of an initial value problem (IVP) for the first-order differential equation with an operator coefficient as a series using the Cayley transform of the corresponding operator coefficient and the Laguerre polynomials. In the case of a boundary value problem (BVP) for the second-order differential equation with an operator coefficient, we represent its solution using the Cayley transform and the Meixner-type polynomials. The approximate solution is the truncated sum of N (the discretization parameter) summands. We give the error estimate of these approximations depending on N and the distance of t to the initial point of the time interval or of the spatial argument x to the boundary of the spatial domain.

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.

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.

We consider an adaptive finite element method with arbitrary but fixed polynomial degree p ≥ 1 {p\geq 1} , where adaptivity is driven by an edge-based residual error estimator. Based on the modified maximum criterion from [L. Diening, C. Kreuzer and R. Stevenson, Instance optimality of the adaptive maximum strategy, Found. Comput. Math. 16 2016, 1, 33–68], we propose a goal-oriented adaptive algorithm and prove that it is instance optimal. More precisely, the goal error is bounded by the product of the total errors (being the sum of energy error plus data oscillations) of the primal and the dual problem, and the proposed algorithm is instance optimal with respect to this upper bound. Numerical experiments underline our theoretical findings.

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.

Consider the Poisson equation in a polyhedral domain with mixed boundary conditions. We establish new regularity results for the solution with possible vertex and edge singularities with interior data in usual Sobolev spaces H σ {H^{\sigma}} with σ ∈ [ 0 , 1 ) {\sigma\in[0,1)} . We propose anisotropic finite element algorithms approximating the singular solution in the optimal convergence rate. In particular, our numerical method involves anisotropic graded meshes with fewer geometric constraints but lacking the maximum angle condition. Optimal convergence on such meshes usually requires the pure Dirichlet boundary condition. Thus, a by-product of our result is to extend the application of these anisotropic meshes to broader practical computations with the price to have “smoother” interior data. Numerical tests validate the theoretical analysis.

We study a system of semi-linear elliptic partial differential equations with a lower order cubic nonlinear term, and inhomogeneous Dirichlet boundary conditions, relevant for two-dimensional bistable liquid crystal devices, within a reduced Landau–de Gennes framework. The main results are (i) a priori error estimates for the energy norm, within the Nitsche’s and discontinuous Galerkin frameworks under milder regularity assumptions on the exact solution and (ii) a reliable and efficient a posteriori analysis for a sufficiently large penalization parameter and a sufficiently fine triangulation in both cases. Numerical examples that validate the theoretical results, are presented separately.

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.

Consider the distributed optimal control problem governed by the von Kármán equations defined on a polygonal domain of ℝ 2 {\mathbb{R}^{2}} that describe the deflection of very thin plates with box constraints on the control variable. This article discusses a numerical approximation of the problem that employs the Morley nonconforming finite element method (FEM) to discretize the state and adjoint variables. The control is discretized using piecewise constants. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Error estimates in lower-order norms for the state and adjoint variables are derived. The lower-order estimates for the adjoint variable and a post-processing of control leads to an improved error estimate for the control variable. Numerical results confirm the theoretical results obtained.