ICOSAHOM 2020
International Conference on Spectral and High Order Methods
12th  16th July 2021  Vienna, Austria
Conference Agenda
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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

Session Overview 
Session  
Z4: preconditioning, spectral methods
 
Presentations  
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 loworder 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 nonzero 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 loworder finite difference discretization, while having a sparser Cholesky factorization. We can thus afford to assemble and factorize the matrices for the vertexpatch problems, even for very high polynomial degree. In the nonseparable 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 BernsteinBezier 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 ^{1}German Aerospace Center, Germany; ^{2}TU 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 tensorproduct 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 loworder solvers, enabling highorder simulations that are as cheap as loworder 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 NavierStokes equations on curvilinear domains ^{1}IIIT Delhi India, India; ^{2}BITS Pilani, Hyderabad Campus Here we present an exponentially accurate least squares formulation based spectral element solver for generalized Stokes equations and NavierStokes 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 NavierStokes 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 HighOrder Spectral ElementLike Method ^{1}Purdue University, West Lafayette, Indiana, USA; ^{2}Purdue University, Fort Wayne, Indiana, USA We present a neural networkbased 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 subdomain is represented by a local feedforward neural network, and C^k continuity conditions are imposed on the subdomain 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 preset 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 backpropagation (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 subdomains) in the system increases. Ample numerical experiments are presented to demonstrate its highorder convergence characteristics and its computational performance in longtime dynamic simulations of timedependent PDEs. 
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