International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
Please note that all times are shown in the time zone of the conference. The current conference time is: 30th Nov 2022, 10:50:21pm CET
Z4: preconditioning, spectral methods
2:00pm - 2:20pm
A sparse preconditioner for $hp$-FEM based on the fast diagonalization method
University of Oxford, United Kingdom
Pavarino proved that the additive Schwarz method with vertex patches and a
low-order coarse space gives a $p$-robust solver for symmetric and coercive
problems. However, for very high polynomial degree it is not feasible to
assemble or factorize the matrices for each patch. In this work we introduce a
direct solver for separable patch problems that scales to very high polynomial
degree on tensor product cells. The solver constructs a tensor product basis
that diagonalizes the blocks in the stiffness matrix for the internal nodes of
each individual cell. As a result, the non-zero structure of the elemental
matrices is that of the graph connecting internal nodes to their corresponding
facet locations. In the new basis, the patch problem is as sparse as a
low-order finite difference discretization, while having a sparser Cholesky
factorization. We can thus afford to assemble and factorize the matrices for
the vertex-patch problems, even for very high polynomial degree. In the
non-separable case, the method can be applied as a preconditioner by
approximating the problem with a separable surrogate.
2:20pm - 2:40pm
High Order H2 Conforming Finite Element Methods on Triangles
Brown University, United States of America
High order numerical methods are known to provide exponential rates of convergence for problems containing singularities and boundary layers. Such features are typical in engineering problems including plates and shells. However, conforming approximation of such problems often requires C1 continuity which, unfortunately, rules out the use of many, if not all, existing finite element codes. We describe the details needed for the efficient implementation of arbitrarily high order finite elements with C1 continuity on unstructured meshes of triangles exploiting properties of the Bernstein-Bezier polynomials. The issue of efficient preconditioning of the resulting systems is addressed. The method is illustrated by applying it to the solution of a number of representative test problems.
2:40pm - 3:00pm
Linearly scaling HDG: The hybridizable interior penalty method
1German Aerospace Center, Germany; 2TU Dresden, Germany
For elliptic equations in three dimensions, the runtime of discontinuous Galerkin methods typically scales with O(p^4 n_e) , where p is the polynomial degree and n_e the number of elements and structure exploitation via tensor-product bases is assumed.
For local discontinuous Galerkin methods that allow for hybridization, the authors were able to linearize the runtime, i.e. to create a solver that scales with O(p^3 n_e) = O (n_DOF), where n_DOF are the total number of degrees of freedom.
This contribution extends the developed techniques to the hybridizable interior penalty method, first deriving a linearly scaling elliptic operator, thereafter a linearly scaling block smoothers and, lastly, combining both into a linearly scaling multigrid method, i.e. O(p^3 n_e) = O (n_DOF).
Runtime tests confirm that the solver requires a similar amount of runtime per degree of freedom as low-order solvers, enabling high-order simulations that are as cheap as low-order simulations.
3:00pm - 3:20pm
Nonstandard Orthogonal Polynomials and Spectral Methods
Eastern Kentycky University, United States of America
We describe constructions of nonstandard orthogonal polyomials, that is, alternative orthogonal polynomials (as well as their rational and exponential analogs) and recently introduced structured orthogonal polynomials. Also, we represent examples on application of nonstandard orthogonal polynomials to solving problems by spectral methods. The examples include problems in computational fluid dynamics, numerical analysis and approximation theory.
3:20pm - 3:40pm
Spectral element solver for generalized Stokes and Navier-Stokes equations on curvilinear domains
1IIIT Delhi India, India; 2BITS Pilani, Hyderabad Campus
Here we present an exponentially accurate least squares formulation based spectral element solver for generalized Stokes equations and Navier-Stokes equations on curvilinear domains. Equal order polynomials are used for both velocity and pressure variables without using any stabilizing parameter. Primitive formulations of generalized Stokes and Navier-Stokes equations are investigated and any first order reformulation has been avoided. A suitable preconditioner has been used to control the iteration numbers of resulting linear system. Preconditioned conjugate gradient method is used to obtain the numerical solution. Numerical experiments with wide range of Reynolds numbers on different curved domains yield similar order accuracy at a cost of higher iteration number with higher Reynolds number. Mass conservation quality of proposed solver is displayed on various curved domains with different Reynolds numbers.
3:40pm - 4:00pm
A Neural Network Based High-Order Spectral Element-Like Method
1Purdue University, West Lafayette, Indiana, USA; 2Purdue University, Fort Wayne, Indiana, USA
We present a neural network-based method for solving linear and nonlinear partial differential equations (PDE) that exhibits convergence characteristics typically found with traditional spectral or spectral element type techniques. This method, termed locELM, combines the ideas of extreme learning machines, domain decomposition, and local neural networks. The field solution on each sub-domain is represented by a local feed-forward neural network, and C^k continuity conditions are imposed on the sub-domain boundaries. Each local neural network may be shallow or deep, consisting of one or multiple hidden layers, while its last hidden layer can be wide. The weight/bias coefficients in the hidden layers are pre-set to random values and fixed, and only the weight coefficients in the output layers of the local neural networks are trainable parameters. The overall neural network is trained by a linear or nonlinear least squares computation, not by the back-propagation (or gradient descent) type algorithms. The current method exhibits a clear sense of convergence with respect to the degrees of freedom in the neural network. Its numerical errors typically decrease exponentially or nearly exponentially as the number of degrees of freedom (e.g. the number of trainable parameters, number of training data points, number of sub-domains) in the system increases. Ample numerical experiments are presented to demonstrate its high-order convergence characteristics and its computational performance in long-time dynamic simulations of time-dependent PDEs.
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