Scalar Elliptic Partial Differential Equations Research Articles

A Stencil Scaling Approach for Accelerating Matrix-Free Finite Element Implementations

We present a novel approach to fast on-the-fly low order finite element assembly for scalar elliptic partial differential equations of Darcy type with variable coefficients optimized for matrix-free implementations. Our approach introduces a new operator that is obtained by appropriately scaling the reference stiffness matrix from the constant coefficient case. Assuming sufficient regularity, an a priori analysis shows that solutions obtained by this approach are unique and have asymptotically optimal order convergence in the $H^1$- and the $L^2$-norms on hierarchical hybrid grids. For the preasymptotic regime, we present a local modification that guarantees uniform ellipticity of the operator. Cost considerations show that our novel approach requires roughly one-third of the floating-point operations compared to a classical finite element assembly scheme employing nodal integration. Our theoretical considerations are illustrated by numerical tests that confirm the expectations with respect to accuracy and run-time. A large scale application with more than a hundred billion ($1.6\cdot10^{11}$) degrees of freedom executed on 14310 compute cores demonstrates the efficiency of the new scaling approach.

Identification problems with given material interfaces

This paper is devoted to the identification of coefficients in scalar elliptic partial differential equations using an optimal control approach. The coefficients usually can be interpreted as material characteristics and play the role of the control variable. The paper focuses on processes in domains which can be split into a finite number of materially homogeneous subdomains, i.e. the coefficients to be identified are piecewise constant on them. In addition, we assume that the material interfaces are a priori known. We prove the existence of at least one solution of the optimal control problem for a large class of cost functionals and show that solutions can be obtained as limits of solutions to the problems which are governed by finite element discretizations of the state equations. Further, the unified algebraic sensitivity analysis of the first and the second order for several least squares type cost functions is investigated. Finally, a model problem of the identification of coefficients characterizing hydraulic conductivity by pumping tests in groundwater flow modeling is numerically solved using the sequential approach and the scalarization technique.

A minimum Sobolev norm technique for the numerical discretization of PDEs

Partial differential equations (PDEs) are discretized into an under-determined system of equations and a minimum Sobolev norm solution is shown to be efficient to compute and converge under very generic conditions. Numerical results of a single code, that can handle PDEs in first-order form on complicated polygonal geometries, are shown for a variety of PDEs: variable coefficient div–curl, scalar elliptic PDEs, elasticity equation, stationary linearized Navier–Stokes, scalar fourth-order elliptic PDEs, telegrapher's equations, singular PDEs, etc.

Algebraic Multigrid for Moderate Order Finite Elements

We investigate the use of algebraic multigrid (AMG) methods for the solution of large sparse linear systems arising from the discretization of scalar elliptic partial differential equations with Lagrangian finite elements of order at most 4. The resulting system matrices do not have the M-matrix property that is required by standard analyses of classical AMG and aggregation-based AMG methods. A unified approach is presented that allows us to extend these analyses. It uses an intermediate M-matrix and highlights the role of the spectral equivalence constant that relates this matrix to the original system matrix. This constant is shown to be bounded independently of the problem size and jumps in the coefficients of the partial differential equations, provided that jumps are located at elements' boundaries. For two-dimensional problems, it is further shown to be uniformly bounded if the angles in the triangulation also satisfy a uniform bound. This analysis validates the application of the AMG methods to the considered problems. On the other hand, because the intermediate M-matrix can be computed automatically, an alternative strategy is to define the AMG preconditioners from this matrix, instead of defining them from the original matrix. Numerical experiments are presented that assess both strategies using publicly available state-of-the-art implementations of classical AMG and aggregation-based AMG methods.

Novel path generation algorithm for high-speed pocket milling

A novel tool path generation algorithm is presented for high-speed pocket milling. A spiral fashion polyline is obtained by linear interpolation which is offset from the pocket boundary. The control points are then inserted into the polyline under exponential regularisation. Finally, the smooth tool path is generated via B-spline curve fitting. The tool path generated through this method can be directly used by the machine controller with a NURBS interpolation function, which avoids the time-consuming computation of dividing them into many short line segments. Compared with conventional methods of direction-parallel and contour-parallel path generation, the tool paths obtained by this algorithm have less cutting force fluctuation because they are Cncontinuous, smooth and have no direction jump. The engineering applications proved that the presented method can not only save machining time over the conventional algorithm, but also can avoid solving time-consuming scalar elliptic partial differential equations which are needed by existing spiral tool path generation method.

Bootstrap AMG

We develop an algebraic multigrid (AMG) setup scheme based on the bootstrap framework for multiscale scientific computation. Our approach uses a weighted least squares definition of interpolation, based on a set of test vectors that are computed by a bootstrap setup cycle and then improved by a multigrid eigensolver and a local residual-based adaptive relaxation process. To emphasize the robustness, efficiency, and flexibility of the individual components of the proposed approach, we include extensive numerical results of the method applied to scalar elliptic partial differential equations discretized on structured meshes. As a first test problem, we consider the Laplace equation discretized on a uniform quadrilateral mesh, a problem for which multigrid is well understood. Then, we consider various more challenging variable coefficient systems coming from covariant finite-difference approximations of the two-dimensional gauge Laplacian system, a commonly used model problem in AMG algorithm development for linear systems arising in lattice field theory computations.

A Nonlocal Vector Calculus with Application to Nonlocal Boundary Value Problems

We develop a calculus for nonlocal operators that mimics Gauss's theorem and Green's identities of the classical vector calculus. The operators we define do not involve derivatives. We then apply the nonlocal calculus to define weak formulations of nonlocal “boundary-value” problems that mimic the Dirichlet and Neumann problems for second-order scalar elliptic partial differential equations. For the nonlocal problems, we derive a fundamental solution and Green's functions, demonstrate that weak formulations of the nonlocal “boundary-value” problems are well posed, and show how, under appropriate limits, the nonlocal problems reduce to their local analogues.

Combinatorial Preconditioners for Scalar Elliptic Finite-Element Problems

We present a new preconditioner for linear systems arising from finite-element discretizations of scalar elliptic partial differential equations (PDE's). The solver splits the collection $\{K_{e}\}$ of element matrices into a subset of matrices that are approximable by diagonally dominant matrices and a subset of matrices that are not approximable. The approximable $K_{e}$'s are approximated by diagonally dominant matrices $L_{e}$'s that are assembled to form a global diagonally dominant matrix L. A combinatorial graph algorithm then approximates L by another diagonally dominant matrix M that is easier to factor. Finally, M is added to the inapproximable elements to form the preconditioner, which is then factored. When all the element matrices are approximable, which is often the case, the preconditioner is provably efficient. Approximating element matrices by diagonally dominant ones is not a new idea, but we present a new approximation method which is both efficient and provably good. The splitting idea is simple and natural in the context of combinatorial preconditioners, but hard to exploit in other preconditioning paradigms. Experimental results show that on problems in which some of the $K_{e}$'s are ill conditioned, our new preconditioner is more effective than an algebraic multigrid solver, than an incomplete-factorization preconditioner, and than a direct solver.

Adjoint pseudospectral least-squares methods for an elliptic boundary value problem

The adjoint approach for Legendre pseudospectral least-squares methods is presented by adopting the adjoint first-order systems developed in [Z. Cai, T. Manteuffel, S. McCormick, J. Ruge, First-order system L L ∗ (FOSLL*): scalar elliptic partial differential equations, SIAM J. Numer. Anal. 39 (2001) 1418–1445]. The discrete adjoint least-squares functional on a polynomial space using Legendre–Gauss–Lobatto points and weights is shown to be equivalent to H 1 norm. The spectral convergence is also provided with several numerical results.

Combinatorial preconditioners for scalar elliptic finite‐element problems

AbstractWe present a new preconditioner for linear systems arising from finite‐elements discretizations of scalar elliptic partial differential equations. The solver is based on building a symmetric diagonally dominant (SDD) approximation of the stiffness matrix K. The approximation is built by approximating each element inside the collection {Ke } of element matrices by an SDD matrix Le. The SDD approximation L is built by assembling the collection {Le }. We then sparsify L using a graph algorithm, and use the sparsified matrix as a preconditioner. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

First-Order System LL* (FOSLL*) for General Scalar Elliptic Problems in the Plane

This paper develops new first-order system LL* (FOSLL*) formulations for scalar elliptic partial differential equations. It extends the work of [Z. Cal et al., SIAM J. Numer. Anal., 39 (2001), pp. 1418--1445], where the FOSLL* methodology was first introduced. One focus of that paper was to develop \FL formulations that allow the use of H1-conforming finite element spaces and optimal multigrid solution techniques to construct L2 approximations of the dependent variables in the presence of discontinuous coefficients. The problems for which this goal was achieved were limited to those with no reaction term and with Dirichlet and Neumann boundaries that were individually connected; that is, each had at most one component. Here, new FOSLL* formulations are developed to achieve the same goals on a wider class of problems, including problems with reaction terms, Dirichlet and Neumann boundaries with multiple components, reentrant corners, and points at which Dirichlet and Neumann boundaries meet with an inner angle greater than $\pi/2$. The efficiency of the improved FOSLL* formulations is illustrated by a series of numerical examples.

First-Order System $\CL\CL^*$ (FOSLL*): Scalar Elliptic Partial Differential Equations

The L2-norm version of first-order system least squares (FOSLS) attempts to reformulate a given system of partial differential equations so that applying a least-squares principle yields a functional whose bilinear part is H1-elliptic. This ellipticity means that the minimization process amounts to solving a weakly coupled system of Poisson-like scalar equations. An unfortunate limitation of the L2-norm FOSLS approach is that this product H1 equivalence generally requires sufficient smoothness of the original problem. Inverse-norm FOSLS overcomes this limitation, but at a substantial loss of real efficiency. The FOSLL* approach introduced here is a promising alternative that is based on recasting the original problem as a minimization principle involving the adjoint equations. This paper provides a theoretical foundation for the FOSLL* methodology and illustrates its performance by applying it numerically to several examples. Results for the so-called two-stage approach applied to discontinuous coefficient problems show promising robustness and optimality. Indeed, FOSLL* appears to exhibit the generality of the inverse-norm FOSLS approach while retaining the full efficiency of the L2-norm approach.