Elliptic Problems Research Articles

On a biharmonic elliptic problem with slightly subcritical non-power nonlinearity

On a biharmonic elliptic problem with slightly subcritical non-power nonlinearity

On a discontinuous nonlinear elliptic problem of nonlocal-type with Neumann boundary condition

On a discontinuous nonlinear elliptic problem of nonlocal-type with Neumann boundary condition

Coupling finite elements of class C1 on composite curved meshes for second order elliptic problems

SummaryFinite elements of class are suitable for the computation of magnetohydrodynamics instabilities in tokamak plasmas. In addition, isoparametric approximations allow for a precise alignment of the mesh with the magnetic field line. Mesh alignment is crucial to achieve axisymmetric equilibria accurately. It is also helpful to deal with the anisotropy nature of magnetized plasma flows. In this numerical framework, several practical simulations are now available. They help to understand better the operation of existing devices and predict the optimal strategies for using the international ITER tokamak under construction. However, a mesh‐aligned isoparametric representation suffers from the presence of critical points of the magnetic field (magnetic axis, X‐point). We here explore a strategy that combines aligned mesh out of the critical points with non‐aligned unstructured mesh in a region containing these points. By this strategy, we can avoid highly stretched elements and the numerical difficulties that come with them. The mesh‐aligned interpolation uses bi‐cubic Hemite‐Bézier polynomials on a structured mesh of curved quadrangular elements. On the other hand, we assume reduced cubic Hsieh‐Clough‐Tocher finite elements on an unstructured triangular mesh. Both meshes overlap, and the resulting formulation is a coupled discrete problem solved iteratively by a suitable one‐level Schwarz algorithm. In this paper, we will focus on the Poisson problem on a two‐dimensional bounded regular domain. This elliptic equation is a simplified version of the axisymmetric tokamak equilibrium one at the asymptotic limit of infinite major radius (large aspect ratio).

Rates of robust superlinear convergence of preconditioned Krylov methods for elliptic FEM problems

AbstractThis paper considers the iterative solution of finite element discretizations of second-order elliptic boundary value problems. Mesh independent estimations are given for the rate of superlinear convergence of preconditioned Krylov methods, involving the connection between the convergence rate and the Lebesgue exponent of the data. Numerical examples demonstrate the theoretical results.

A class of elliptic problems driven by a mixed local-nonlocal operator

In this paper, we consider a class of elliptic problems driven by a mixed local-nonlocal operator. By estimating the critical groups and using Morse theory, the existence of nontrivial solutions is obtained in different cases. Our results extend some classical theorems for semilinear elliptic equations to the mixed local-nonlocal setting.

An overlapping domain decomposition method for the solution of parametric elliptic problems via proper generalized decomposition

A non-intrusive proper generalized decomposition (PGD) strategy, coupled with an overlapping domain decomposition (DD) method, is proposed to efficiently construct surrogate models of parametric linear elliptic problems. A parametric multi-domain formulation is presented, with local subproblems featuring arbitrary Dirichlet interface conditions represented through the traces of the finite element functions used for spatial discretization at the subdomain level, with no need for additional auxiliary basis functions. The linearity of the operator is exploited to devise low-dimensional problems with only few active boundary parameters. An overlapping Schwarz method is used to glue the local surrogate models, solving a linear system for the nodal values of the parametric solution at the interfaces, without introducing Lagrange multipliers to enforce the continuity in the overlapping region. The proposed DD-PGD methodology relies on a fully algebraic formulation allowing for real-time computation based on the efficient interpolation of the local surrogate models in the parametric space, with no additional problems to be solved during the execution of the Schwarz algorithm. Numerical results for parametric diffusion and convection–diffusion problems are presented to showcase the accuracy of the DD-PGD approach, its robustness in different regimes and its superior performance with respect to standard high-fidelity DD methods.

Wolff potentials and local behavior of solutions to elliptic problems with Orlicz growth and measure data

Abstract We establish pointwise bounds expressed in terms of a nonlinear potential of a generalized Wolff type for 𝒜 {{\mathcal{A}}} -superharmonic functions with nonlinear operator 𝒜 : Ω × ℝ n → ℝ n {{\mathcal{A}}:\Omega\times{\mathbb{R}^{n}}\to{\mathbb{R}^{n}}} having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls estimates from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfy conditions expressed in the natural scales. Finally, we give a variant of Hedberg–Wolff theorem on characterization of the dual of the Orlicz space.

Sign-changing solution for an overdetermined elliptic problem on unbounded domain

Abstract We prove the existence of two smooth families of unbounded domains in ℝ N + 1 {\mathbb{R}^{N+1}} with N ≥ 1 {N\geq 1} such that { - Δ ⁢ u = λ ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , ∂ ν ⁡ u = const on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}{-}\Delta u&\displaystyle=\lambda u&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\\ \displaystyle\partial_{\nu}u&\displaystyle=\mathrm{const}&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right. admits a sign-changing solution. The domains bifurcate from the straight cylinder B 1 × ℝ {B_{1}\times\mathbb{R}} , where B 1 {B_{1}} is the unit ball in ℝ N {\mathbb{R}^{N}} . These results can be regarded as counterexamples to the Berenstein Conjecture on unbounded domain. Unlike most previous papers in this direction, a very delicate issue here is that there may be two-dimensional kernel space at some bifurcation point. Thus a Crandall–Rabinowitz-type bifurcation theorem from high-dimensional kernel space is also established to achieve the goal.

Energy estimate up to the boundary for stable solutions to semilinear elliptic problems

We obtain a universal energy estimate up to the boundary for stable solutions of semilinear equations with variable coefficients. Namely, we consider solutions to −Lu=f(u), where L is a linear uniformly elliptic operator and f is C1, such that the linearized equation −L−f′(u) has nonnegative principal eigenvalue. Our main result is an estimate for the L2+γ norm of the gradient of stable solutions vanishing on the flat part of a half-ball, for any nonnegative and nondecreasing f. This bound only requires the elliptic coefficients to be Lipschitz. As a consequence, our estimate continues to hold in general C1,1 domains if we further assume the nonlinearity f to be convex. This result is new even for the Laplacian, for which a C3 regularity assumption on the domain was needed.

Weak Galerkin finite element methods for H(curl;Ω) and H(curl,div;Ω)-elliptic problems

Weak Galerkin finite element methods for H(curl;Ω) and H(curl,div;Ω)-elliptic problems

A two-level additive Schwarz preconditioner for the Nitsche extended finite element approximation of elliptic interface problems

A two-level additive Schwarz preconditioner for the Nitsche extended finite element approximation of elliptic interface problems

On solving elliptic obstacle problems by constant abs-linearization

We consider optimal control problems governed by an elliptic variational inequality of the first kind, namely the obstacle problem. The variational inequality is treated by penalization, which leads to optimization problems governed by a nonsmooth semi-linear elliptic PDE. The CALi algorithm is then applied for the efficient solution of these nonsmooth optimization problems. The special feature of the optimization algorithm CALi is the treatment of the nonsmooth Lipschitz-continuous operators abs, max and min, which allows to explicitly exploit the nonsmooth structure. Stationary points are located by appropriate decomposition of the optimization problem into so-called smooth constant abs-linearized problems. Each of these constant abs-linearized problems can be solved by classical means. The comprehensive algorithmic concept is presented, and its performance is discussed through examples.

An immersed weak Galerkin method for elliptic interface problems on polygonal meshes

An immersed weak Galerkin method for elliptic interface problems on polygonal meshes

A diagonal finite element-projection-proximal gradient algorithm for elliptic optimal control problem

A diagonal finite element-projection-proximal gradient algorithm for elliptic optimal control problem

Multilevel Monte Carlo FEM for elliptic PDEs with Besov random tree priors

AbstractWe develop a multilevel Monte Carlo (MLMC)-FEM algorithm for linear, elliptic diffusion problems in polytopal domain$${\mathcal {D}}\subset {\mathbb {R}}^d$$D⊂Rd, with Besov-tree random coefficients. This is to say that the logarithms of the diffusion coefficients are sampled from so-called Besov-tree priors, which have recently been proposed to model data for fractal phenomena in science and engineering. Numerical analysis of the fully discrete FEM for the elliptic PDE includes quadrature approximation and must account for (a) nonuniform pathwise upper and lower coefficient bounds, and for (b) low path-regularity of the Besov-tree coefficients. Admissible non-parametric random coefficients correspond to random functions exhibiting singularities on random fractals with tunable fractal dimension, but involve no a-priori specification of the fractal geometry of singular supports of sample paths. Optimal complexity and convergence rate estimates for quantities of interest and for their second moments are proved. A convergence analysis for MLMC-FEM is performed which yields choices of the algorithmic steering parameters for efficient implementation. A complexity (“error vs work”) analysis of the MLMC-FEM approximations is provided.

Preconditioners based on matrix splitting for the structured systems from elliptic PDE-constrained optimization problems

With regard to the structured systems of linear equations, which arises from the Galerkin finite element discretizations of elliptic PDE-constrained optimization problems, some new preconditioners based on coefficient matrix splitting are established to speed up the convergence rate of Krylov subspace methods such as GMRES. Additionally, eigenvalue distribution of the corresponding preconditioned matrices is deeply discussed. Numerical simulations are carried out, the results of which exhibit that the corresponding preconditioned GMRES methods perform very well with the theoretical analysis results and are superior to other newly devised preconditioners.

Fractional differentiability for elliptic measure data problems with variable exponents

We investigate quasilinear elliptic equations with variable growth exponents having a Borel measure on the right-hand side. We prove fractional differentiability results for a certain nonlinear expression of the gradient of solutions.

Self-improving properties of very weak solutions to double phase systems

We prove the self-improving property of very weak solutions to non-uniformly elliptic problems of double phase type in divergence form under sharp assumptions on the nonlinearity.

Nonuniformly elliptic Schauder theory

Local Schauder theory holds in the nonuniformly elliptic setting. Specifically, first derivatives of solutions to nonuniformly elliptic problems are locally Hölder continuous if so are their coefficients.

A New Hybrid Block Method for Solving First-Order Differential System Models in Applied Sciences and Engineering

This paper presents a new hybrid block method formulated in variable stepsize mode to solve some first-order initial value problems of ODEs and time-dependent partial differential equations in applied sciences and engineering. The proposed method is implemented considering an adaptive stepsize strategy to maintain the estimated error in each step within a specified tolerance. In order to evaluate the performance and usefulness of the proposed technique in real-world applications, several differential problems from applied sciences and engineering, such as the SIR model, Jacobi elliptic function problem, and chemical reactions problems, are solved numerically. The results of numerical simulations in this work demonstrate that the proposed method is more efficient than other existing numerical methods used for comparisons.