Scalar PDEs Research Articles

Optimal local truncation error method to solution of 2‐D time‐independent elasticity problems with optimal accuracy on irregular domains and unfitted Cartesian meshes

AbstractRecently we have developed the optimal local truncation error method (OLTEM) for scalar PDEs on irregular domains and unfitted Cartesian meshes. Here, OLTEM is extended to a much more general case of a system of PDEs for the 2‐D time‐independent elasticity equations on irregular domains. Compact 9‐point uniform and nonuniform stencils (with the computational costs of linear finite elements) are used with OLTEM. The stencil coefficients are assumed to be unknown and are calculated by the minimization of the local truncation error. It is shown that the second order of accuracy is the maximum possible accuracy for 9‐point stencils independent of the numerical technique used for their derivations. The special treatment of the Neumann boundary conditions has been developed that does not increase the size of the stencils. The numerical examples are in agreement with the theoretical findings. They also show that due to the minimization of the local truncation error, OLTEM is much more accurate than linear finite elements and than quadratic finite elements (up to engineering accuracy of 0.1%–1%) at the same numbers of degrees of freedom. Due to the computational efficiency and trivial unfitted Cartesian meshes that are independent of irregular domains, the proposed technique with no remeshing for the shape change of irregular domains will be effective for many engineering applications.

Non-autonomous scalar linear-dissipative and purely dissipative parabolic PDEs over a compact base flow

In this paper a family of non-autonomous scalar parabolic PDEs over a general compact and connected flow is considered. The existence or not of a neighbourhood of zero where the problems are linear has an influence on the methods used and on the dynamics of the induced skew-product semiflow. That is why two cases are distinguished: linear-dissipative and purely dissipative problems. In both cases, the structure of the global and pullback attractors is studied using principal spectral theory. Besides, in the purely dissipative setting, a simple condition is given, involving both the underlying linear dynamics and some properties of the nonlinear term, to determine the nontrivial sections of the attractor.

Asymptotic Expansions for High-Contrast Scalar and Vectorial PDEs

This paper is about solutions to PDEs in which the ratio between the largest and smallest values of the coefficient is very large. Such problems are referred to as high-contrast ones. A full asympt...

Solutions of Second-Order PDEs with First-Order Quotients

We investigate a general approach for solving a partial differential equation by using the differential invariants of its point symmetries. By first solving its quotient PDE, which is given by the differential syzygies in the algebra of differential invariants, we obtain new differential constraints which are compatible with the PDE under consideration. Adding these constraints to our system makes it overdetermined, and easier to solve. We focus on second-order scalar PDEs on a function of two variables whose quotients are first-order scalar PDEs. This situation occurs only when the Lie algebra of symmetries of the second-order PDE is infinite-dimensional. We apply this idea to various second-order PDEs with infinite-dimensional symmetry Lie algebras, one of which is the Hunter-Saxton equation.

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In [13], for any (1+1)-dimensional scalar evolution equation E, we defined a family of Lie algebras F(E) which are responsible for all ZCRs of E in the following sense. Representations of the algebras F(E) classify all ZCRs of the equation E up to local gauge transformations. In [12] we showed that, using these algebras, one obtains necessary conditions for existence of a Bäcklund transformation between two given equations.In this paper we show that, using the algebras F(E), one obtains some necessary conditions for integrability of (1+1)-dimensional scalar evolution PDEs, where integrability is understood in the sense of soliton theory. Using these conditions, we prove non-integrability for some scalar evolution PDEs of order 5. Also, we prove a result announced in [13] on the structure of the algebras F(E) for certain classes of equations of orders 3, 5, 7, which include KdV, mKdV, Kaup–Kupershmidt, Sawada–Kotera type equations. Among the obtained algebras for equations considered in this paper and in [12], one finds infinite-dimensional Lie algebras of certain polynomial matrix-valued functions on affine algebraic curves of genus 1 and 0.

Domain decomposition preconditioners for multiscale problems in linear elasticity

SummaryWe analyze two‐level overlapping Schwarz domain decomposition methods for vector‐valued piecewise linear finite element discretizations of the PDE system of linear elasticity. The focus of our study lies in the application to compressible, particle‐reinforced composites in 3D with large jumps in their material coefficients. We present coefficient‐explicit bounds for the condition number of the two‐level additive Schwarz preconditioned linear system. Thereby, we do not require that the coefficients are resolved by the coarse mesh. The bounds show a dependence of the condition number on the energy of the coarse basis functions, the coarse mesh, and the overlap parameters, as well as the coefficient variation. Similar estimates have been developed for scalar elliptic PDEs by Graham et al. The coarse spaces to which they apply here are assumed to contain the rigid body modes and can be considered as generalizations of the space of piecewise linear vector‐valued functions on a coarse triangulation. The developed estimates provide a concept for the construction of coarse spaces, which can lead to preconditioners that are robust with respect to high contrasts in Young's modulus and the Poisson ratio of the underlying composite. To confirm the sharpness of the theoretical findings, we present numerical results in 3D using vector‐valued linear, multiscale finite element and energy‐minimizing coarse spaces. The theory is not restricted to the isotropic (Lamé) case, extends to the full‐tensor case, and allows applications to multiphase materials with anisotropic constituents in two and three spatial dimensions. However, the bounds will depend on the ratio of largest to smallest eigenvalue of the elasticity tensor.

Contact manifolds, Lagrangian Grassmannians and PDEs

Abstract In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n + 1)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a Ph.D course given by two of the authors (G. M. and G. M.). As such, it was mainly designed as a quick introduction to the subject for graduate students. But also the more demanding reader will be gratified, thanks to the frequent references to current research topics and glimpses of higher-level mathematics, found mostly in the last sections.

The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs

In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimensional manifold in a natural way. The formal solution space of the nonlinear PDE then is a subspace of this jet space, and inherits from it the structure of a profinite dimensional manifold, if the PDE is formally integrable. We apply our concept to scalar PDEs and prove a new criterion for formal integrability of such PDEs. In particular, this result entails that the Euler-Lagrange equation of a relativistic scalar field with a polynomial self-interaction is formally integrable.

Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes

High order strong stability preserving (SSP) time discretizations are advantageous for use with spatial discretizations with nonlinear stability properties for the solution of hyperbolic PDEs. The search for high order strong stability time-stepping methods with large allowable strong stability time-step has been an active area of research over the last two decades. Recently, multiderivative time-stepping methods have been implemented with hyperbolic PDEs. In this work we describe sufficient conditions for a two-derivative multistage method to be SSP, and find some optimal SSP multistage two-derivative methods. While explicit SSP Runge---Kutta methods exist only up to fourth order, we show that this order barrier is broken for explicit multi-stage two-derivative methods by designing a three stage fifth order SSP method. These methods are tested on simple scalar PDEs to verify the order of convergence, and demonstrate the need for the SSP condition and the sharpness of the SSP time-step in many cases.

A massively parallel solver for discrete Poisson-like problems

The paper considers the parallel implementation of an algebraic multigrid method. The sequential version is well suited to solve linear systems arising from the discretization of scalar elliptic PDEs. It is scalable in the sense that the time needed to solve a system is (under known conditions) proportional to the number of unknowns. The associate software code is also robust and often significantly faster than other algebraic multigrid solvers. The present work addresses the challenge of porting it on massively parallel computers. In this view, some critical components are redesigned, in a relatively simple yet not straightforward way. Thanks to this, excellent weak scalability results are obtained on three petascale machines among the most powerful today available.

Self-similar solutions for active scalar equations in Fourier–Besov–Morrey spaces

We are concerned with a family of dissipative active scalar equation with velocity fields coupled via multiplier operators that can be of positive-order. We consider sub-critical values for the fractional diffusion and prove global well-posedness of solutions with small initial data belonging to a framework based on Fourier transform, namely Fourier–Besov–Morrey spaces. Since the smallness condition is with respect to the weak norm of this space, some initial data with large \(L^{2}\)-norm can be considered. Self-similar solutions are obtained depending on the homogeneity of the initial data and couplings. Also, we show that solutions are asymptotically self-similar at infinity. Our results can be applied in a unified way for a number of active scalar PDEs like 1D models on dislocation dynamics in crystals, Burgers’ equation, 2D vorticity equation, 2D generalized SQG, 3D magneto-geostrophic equations, among others.

Decoupled quantum walks, models of the Klein-Gordon and wave equations

Decoupling a vectorial PDE consists in solving the system for each component, thereby obtaining scalar PDEs that prescribe the evolution of each component independently. We present a general approach to decoupling of Quantum Walks, again defined as a procedure to obtain an evolution law for each scalar component of the Quantum Walks, in such a way that it does not depend on the other components. In particular, the method is applied to show the relation between the Dirac (or Weyl) Quantum Walk in three space dimensions with (or without) mass term, and the Klein-Gordon (or wave) equation.

Generalized Lie-Bäcklund theorem for Lie class $$\omega =1$$ ω = 1 overdetermined systems

In this paper we prove a version of Lie-Backlund theorem for overdetermined systems of scalar PDEs, whose general solution depends on 1 function of 1 variable. This generalizes the case of involutive system of the second order on the plane treated by E. Cartan in 1910. The approach is based on the geometric theory of PDEs and Tanaka theory. Many examples are provided.

Multiscale finite element coarse spaces for the application to linear elasticity

Abstract We extend the multiscale finite element method (MsFEM) as formulated by Hou and Wu in [Hou T.Y., Wu X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 1997, 134(1), 169–189] to the PDE system of linear elasticity. The application, motivated by the multiscale analysis of highly heterogeneous composite materials, is twofold. Resolving the heterogeneities on the finest scale, we utilize the linear MsFEM basis for the construction of robust coarse spaces in the context of two-level overlapping domain decomposition preconditioners. We motivate and explain the construction and show that the constructed multiscale coarse space contains all the rigid body modes. Under the assumption that the material jumps are isolated, that is they occur only in the interior of the coarse grid elements, our numerical experiments show uniform convergence rates independent of the contrast in Young’s modulus within the heterogeneous material. Elsewise, if no restrictions on the position of the high coefficient inclusions are imposed, robustness cannot be guaranteed any more. These results justify expectations to obtain coefficient-explicit condition number bounds for the PDE system of linear elasticity similar to existing ones for scalar elliptic PDEs as given in the work of Graham, Lechner and Scheichl [Graham I.G., Lechner P.O., Scheichl R., Domain decomposition for multiscale PDEs, Numer. Math., 2007, 106(4), 589–626]. Furthermore, we numerically observe the properties of the MsFEM coarse space for linear elasticity in an upscaling framework. Therefore, we present experimental results showing the approximation errors of the multiscale coarse space w.r.t. the fine-scale solution.

LAPLACE TRANSFORMATION OF LIE CLASS ω = 1 OVERDETERMINED SYSTEMS

In this paper, we investigate overdetermined systems of scalar PDEs on the plane with one common characteristic, whose general solution depends on one function of one variable. We describe linearization of such systems and their integration via Laplace transformation, relating this to Lie's integration theorem and formal theory of PDEs.

General Constrained Energy Minimization Interpolation Mappings for AMG

This report proposes a new class of interpolation procedures for use in algebraic multigrid (AMG). The procedure is general in that it applies to s.p.d. finite element discretization problems that include scalar and vector elliptic PDEs, and it can be adapted to the time-domain Maxwell (definite $H(\mathrm{curl})$) problems. It can also be viewed as an extension of previously proposed vector-fitting interpolation procedures and can be used in adaptive AMG cycles. We illustrate the performance of the new interpolation matrices on a number of test problems.

Towards Adaptive Smoothed Aggregation ($\alpha$SA) for Nonsymmetric Problems

Applying smoothed aggregation (SA) multigrid to solve a nonsymmetric linear system, $A\mathbf{x} =\mathbf{b}$, is often impeded by the lack of a minimization principle that can be used as a basis for the coarse-grid correction process. This paper proposes a Petrov-Galerkin (PG) approach based on applying SA to either of two symmetric positive definite (SPD) matrices, $\sqrt{A^{t}A}$ or $\sqrt{AA^{t}}$. These matrices, however, are typically full and difficult to compute, so it is not computationally efficient to use them directly to form a coarse-grid correction. The proposed approach approximates these coarse-grid corrections by using SA to accurately approximate the right and left singular vectors of $A$ that correspond to the lowest singular value. These left and right singular vectors are used to construct the restriction and interpolation operators, respectively. A preliminary two-level convergence theory is presented, suggesting that more relaxation should be applied than for an SPD problem. Additionally, a nonsymmetric version of adaptive SA ($\alpha$SA) is given that automatically constructs SA multigrid hierarchies using a stationary relaxation process on all levels. Numerical results are reported for convection-diffusion problems in two dimensions with varying amounts of convection for constant, variable, and recirculating convection fields. The results suggest that the proposed approach is algorithmically scalable for problems coming from these nonsymmetric scalar PDEs (with the exception of recirculating flow). This paper serves as a first step for nonsymmetric $\alpha$SA. The long-term goal of this effort is to develop nonsymmetric $\alpha$SA for systems of PDEs, where the SA framework has proven to be well suited for adaptivity in SPD problems.

A multiresolution space‐time adaptive scheme for the bidomain model in electrocardiology

AbstractThe bidomain model of electrical activity of myocardial tissue consists of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time‐dependent ODE modeling the evolution of the gating variable. In the simpler subcase of the monodomain model, the elliptic PDE reduces to an algebraic equation. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge‐Kutta‐Fehlberg‐type adaptive time integration. A series of numerical examples demonstrates that these methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, the optimal choice of the threshold for discarding nonsignificant information in the multiresolution representation of the solution is addressed, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed‐up, memory compression, and errors in different norms. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

On classification of second-order PDEs possessing partner symmetries

Recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of the heavenly equations of Plebañski that govern heavenly gravitational metrics. In this paper, we present a classification of scalar second-order PDEs with four variables that possess partner symmetries and contain only second derivatives of the unknown. We present a general form of such a PDE together with recursion relations between partner symmetries. This general PDE is transformed to several simplest canonical forms containing two heavenly equations of Plebañski among them and two other nonlinear equations which we call the mixed heavenly equation and asymmetric heavenly equation. We have calculated all the point and contact symmetries of all the canonical equations which can be used as an input in our recursion relations. On an example of the mixed heavenly equation, we show how to use partner symmetries for obtaining noninvariant solutions of PDEs by a lift from invariant solutions. Finally, we present Ricci-flat self-dual metrics governed by solutions of the mixed heavenly equation and its Legendre transform.

Scalar parabolic PDEs and braids

The comparison principle for scalar second order parabolic PDEs on functions u ( t , x ) u(t,x) admits a topological interpretation: pairs of solutions, u 1 ( t , ⋅ ) u^1(t,\cdot ) and u 2 ( t , ⋅ ) u^2(t,\cdot ) , evolve so as to not increase the intersection number of their graphs. We generalize to the case of multiple solutions { u α ( t , ⋅ ) } α = 1 n \{u^\alpha (t,\cdot )\}_{\alpha =1}^n . By lifting the graphs to Legendrian braids, we give a global version of the comparison principle: the curves u α ( t , ⋅ ) u^\alpha (t,\cdot ) evolve so as to (weakly) decrease the algebraic length of the braid. We define a Morse-type theory on Legendrian braids which we demonstrate is useful for detecting stationary and periodic solutions to scalar parabolic PDEs. This is done via discretization to a finite-dimensional system and a suitable Conley index for discrete braids. The result is a toolbox of purely topological methods for finding invariant sets of scalar parabolic PDEs. We give several examples of spatially inhomogeneous systems possessing infinite collections of intricate stationary and time-periodic solutions.