Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1. Selected Papers from the ICOSAHOM Conference, Vienna, Austria, July 12-16, 2021

Preface
Contents
Part I Invited Papers
	On Arbitrarily Regular Conforming Virtual Element Methods for Elliptic Partial Differential Equations
		1 Introduction
		2 The Continuous Problem
		3 The Discrete Problem and an Abstract Convergence Result
			3.1 Mesh Notation, Mesh Regularity and Some Basic Definitions
			3.2 Abstract Convergence Theorem
		4 The Virtual Element Spaces of Higher-Order Continuity
			4.1 Local Space Definitions
			4.2 Local Degrees of Freedom
			4.3 Global Virtual Element Spaces
			4.4 Elliptic Projection Operator
			4.5 Enhancement
			4.6 The Virtual Element Bilinear Form ah(·,·)
			4.7 The Virtual Element Approximation of the Load Term
			4.8 Error Analysis
		5 Conclusion
		References
	Broadband Recursive Skeletonization
		1 Introduction
		2 Preliminaries
			2.1 Integral Equations of Scattering Theory
			2.2 Nyström Discretization
			2.3 Interpolative Decomposition
			2.4 Recursive Skeletonization
		3 Theoretical Foundations
			3.1 Universal Bases
			3.2 Broadband Analysis on Concentric Circles
			3.3 Broadband Analysis in 2D
			3.4 Broadband Analysis in 3D
		4 Connection with Fourier Operators
		5 Algorithms
		6 Broadband Recursive Skeletonization
		7 Applications and Numerical Experiments
			7.1 2D Boundary Integral Equation
			7.2 2D Eigenvalue Problems
			7.3 3D Boundary Integral Equation
		8 Conclusions and Future Work
		References
	A Novel Spectral Method for the Subdiffusion Equation
		1 Introduction
		2 Functional Spaces
			2.1 ψ-Fractional Operators and Variable Transformation
			2.2 ψ-Fractional Sobolev Spaces
			2.3 Some Useful Lemmas
		3 A Spectral Method for Fractional Ordinary Differential Equations
			3.1 Well-Posedness
			3.2 Error Estimate
			3.3 Implementation
		4  Application to the Time Fractional Subdiffusion Equations
		5 Numerical Examples
		6 Concluding Remarks
		References
Part II Contributed Papers
	A Face-Based Eight-Order Scheme for Convection-Diffusion Problems with Polyhedral Unstructured Grids
		1 Introduction
		2 Finite Volume Discretization in 1D Grids
			2.1 Discretization of the Face Value and Gradient
			2.2 Contributions from the Boundaries
			2.3 The Peclet Number
			2.4 Tested Grids in 1D Results
		3 Implementation Verification with 1D Grids
		4 Results with Polyhedral Unstructured Grids
		5 Conclusions
		References
	Flexible Weights for High Order Face Based Finite ElementInterpolation
		1 Introduction
		2 Distributions of Unisolvent and Minimal 2-Simplices
			2.1 With Overlapping
			2.2 With Holes
		3 Towards Non-uniform Distributions of Small Faces
		4 Numerical Results
		5 Conclusions
		Reference
	Taming the CFL Number for Discontinuous Galerkin Methods by Local Exponentiation
		1 Introduction
		2 Time-Stepping by Local Approximation to the Matrix Exponential
			2.1 Local Exponentiation
			2.2 Exponential Locality and Decay of Discretizations of Hyperbolic Operators
			2.3 Cost and Complexity for Local Exponentiation
		3 Numerical Experiments in One Dimension: Maxwell\'s Equations
			3.1 Comparison of Spectra
			3.2 Convergence Properties of the Method
		4 Conclusion
		References
	hp-Finite Elements with Decoupled Constraints for Elastoplasticity
		1 Introduction
		2 The Model Problem and Its hp-FE Discretization
		3 Representation as Nonlinear Equation
		4 Numerical Examples
		References
	Convergence of Finite Difference Schemes: Matrix Versus Kernel Analysis
		1 Biharmonic Time Dependent Problem, Continous and Semidiscrete
		2 Error Analysis for the Discrete Biharmonic Operator (DBO)
		3 Convergence of the Discrete Time-Evolution Solution by Purely Matrix Method
		4 Convergence of the Discrete Time-Evolution Solution by Kernel Analysis
		5 Comments and Perspectives
		References
	Shape Optimization with Nonlinear Conjugate Gradient Methods
		1 Introduction
		2 Theoretical Background
			2.1 Fundamentals of Shape Optimization
			2.2 Shape Calculus
			2.3 A Riemannian View on Shape Optimization and Steklov-Poincaré-Type Metrics
		3 Nonlinear CG Methods for Shape Optimization
		4 Numerical Examples
			4.1 Shape Optimization with a Poisson Equation
			4.2 Shape Optimization of Energy Dissipation in a Pipe
		5 Conclusions
		References
	SpecSolve: Spectral Methods for Spectral Measures
		1 Introduction
		2 Spectral Measures
		3 Algorithmic Framework for SpecSolve
		4 Singular Integro-Differential Operators
		5 Linear Operator Pencils
			5.1 Recovering a Self-Adjoint Operator
			5.2 Framework for Generalized Spectral Measures
			5.3 Examples
		References
	Assessment of a Wall Distance Free Transition Model Based on the Laminar Kinetic Energy in a Discontinuous Galerkin Solver
		1 Introduction
		2 Model
		3 Numerical Framework
		4 Results
		5 Conclusions
		References
	A Posteriori Error Estimate and Mesh Adaptation for the Numerical Solution of the Richards Equation
		1 Introduction
		2 Continuous Problem
		3 STDGM Discretizations
		4 Error Measure
		5 Error Estimates
			5.1 Temporal Flux Reconstruction of the Solution
			5.2 Spatial Flux Reconstruction of the Solution
			5.3 Upper Bound
			5.4 Lower Bound
		6 Rates of Convergence of the Error Estimator
		7 Simulation of the Single Ring Infiltration Process
		References
	A Data-Driven Partitioned Approach for the Resolution of Time-Dependent Optimal Control Problems with Dynamic Mode Decomposition
		1 Introduction
		2 Time-Dependent Optimal Control Problems
		3 Dynamic Mode Decomposition with Control
		4 A Partitioned Approach for Time-Dependent OCPs
		5 Numerical Results
			5.1 Boundary OCP Governed by a Graetz Flow
			5.2 OCP Governed by Time-Dependent Stokes Equations
			5.3 Speedup Considerations
		6 Conclusions and Perspectives
		References
	A New Multiscale Discontinuous Galerkin Method for a Class of Second-Order Equations with Oscillatory Solutions in Two-Dimensional Space
		1 Introduction
		2 Multiscale DG Method
		3 Numerical Results
			3.1 Constant f and ω=f
			3.2 Constant f, ω≠f
			3.3 Applications to 2D Schrödinger Equation
		4 Concluding Remarks
		References
	Hybrid High-Order Methods for Elliptic PDEs on Curved and Complicated Domains
		1 Introduction
		2 Model and Discrete Setting
			2.1 Model Problem
			2.2 Polytopal and Curved Meshes
			2.3 Analysis Tools
		3 HHO Discretization
		4 Main Results
			4.1 Stability and Well-Posedness
			4.2 Approximation and H1 Error Estimate
		5 Numerical Experiment
		References
	On Higher Order Passivity Preserving Schemes for Nonlinear Maxwell\'s Equations
		1 Introduction
		2 The e–h Formulation
		3 Discretization of the e–h Formulation
		4 The e–a Formulation
		5 Discretization of e-a Formulation
		6 Numerical Validation
			6.1 Results for the e–h Formulation
			6.2 Numerical Results for the e–a Formulation
		7 Discussion
		References
	Electromagnetic Displacements Rotating Inside an Annular Region
		1 Introduction
		2 Preliminaries
		3 Extension on the Equatorial Plane Outside the Sphere
		4 Determination of the Parameters
		5 Final Comments and Speculations
		References
	Exponential Convergence of hp-FEM for the Integral Fractional Laplacian in 1D
		1 Introduction
		2 Model Problem and Main Results
			2.1 The Fractional Laplacian
			2.2 Weighted Analytic Regularity
			2.3 Exponential Convergence of hp-FEM
		3 Regularity Results
			3.1 The Caffarelli-Silvestre Extension
			3.2 Global Regularity
			3.3 Interior Regularity
			3.4 Exponential Convergence of hp-FEM
		4 Numerical Example
		References
	Discontinuous Galerkin Method for Linear Wave Equations Involving Derivatives of the Dirac Delta Distribution
		1 Introduction
		2 Reduction to a First-Order System
			2.1 Removing Singular Behavior
			2.2 First-Order Reduction
			2.3 Distributional Constraint
		3 Discontinuous Galerkin Method
			3.1 The Source-Free Method
			3.2 Modifications to the Numerical Flux
		4 Distributional Solutions to the 1+1 Wave Equation
		5 Numerical Results
			5.1 Wave Equation with a δ(2)(x) Source Term
			5.2 Persistent Spurious Solutions from Distributional Constraint Violations
		6 Final Remarks
		References
	An Energy-Preserving High Order Method for Liouville\'s Equation of Geometrical Optics
		1 Introduction
		2 Liouville\'s Equation
		3 Numerical Method
			3.1 Space Discretisation
			3.2 Optical Interfaces
		4 Results
			4.1 Energy-Preserving and Convergence
			4.2 Comparison with Ray Tracing
		5 Conclusion
		References
	Using 1-Regularization for Shock Capturing in Discontinuous Galerkin Methods
		1 Introduction
		2 1-Regularization and Polynomial Annihilation
			2.1 1-Regularization
			2.2 Connection to Numerical Hyperbolic Conservation Laws
			2.3 Polynomial Annihilation
		3 Shock Capturing by 1-Regularization
			3.1 Proposed Procedure
			3.2 Parameter Selection
		4 Preserving Local Conservation
		5 Numerical Tests
		6 Summary
		References
	Parallel Simulations of High-Power Optical Fiber Amplifiers
		1 Introduction
		2 Fiber Amplifier Model
			2.1 Ytterbium-Doped Fiber Amplifier
			2.2 DPG Finite Element Discretization
		3 Numerical Results
			3.1 Computational Performance
			3.2 Large-Scale Fiber Simulations
		4 Conclusions
		References
	Reduced Order Modeling for Spectral Element Methods: Current Developments in Nektar++ and Further Perspectives
		1 Introduction
		2 Continuous Galerkin: Incompressible Flow Simulations
			2.1 Overview
			2.2 Reduced Order Modeling for Continuous Galerkin Methods
				2.2.1 A Numerical Example
				2.2.2 Parametric Variation in Geometry
		3 Discontinuous Galerkin: Compressible Flow Simulations
			3.1 Overview
			3.2 Reduced Order Modeling for Discontinuous Galerkin Methods
			3.3 Results
		4 Conclusions
		References
	Interface Discontinuities in Spectral-Element Simulations with Adaptive Mesh Refinement
		1 Introduction
		2 Problem Description
			2.1 Turbulent Pipe Flow
		3 Numerical Tests and Results
			3.1 Manual Refinement
			3.2 Error-Driven Refinement
			3.3 Predictive Model
		4 Conclusions
		References
	Fully Discrete Entropy Conserving/Stable Discontinuous Galerkin Discretization of the Euler Equations in Entropy Variables
		1 Introduction
		2 Spatial and Temporal Discretization
		3 Numerical Results
			3.1 The Isentropic Vortex Convection Problem
			3.2 The Inviscid Double Shear Layer
		4 Conclusions
		References
	Inexact IETI-DP for Conforming Isogeometric Multi-Patch Discretizations
		1 Introduction
		2 Model Problem and Its Solution
		3 Analysis of the Method
			3.1 Analysis of the Main Iteration
			3.2 Analysis of the System for the Primal Basis
		4 Numerical Results
		References
	Split Form ALE DG Methods for the Euler Equations: Entropy Stability and Kinetic Energy Dissipation
		1 Introduction
		2 The Euler Equations
			2.1 The ALE Transformation
			2.2 Entropy Analysis in Three Dimensions
			2.3 Analysis of the Kinetic Energy Evolution
		3 Discontinuous Galerkin Spectral Element Method
			3.1 Building Blocks for the Spectral Element Approximation
				3.1.1 Discrete Integrals
				3.1.2 Discrete Metric Identities
			3.2 Standard ALE DGSEM
			3.3 Split Form ALE DGSEM
			3.4 EC/KEP Split form ALE DGSEM
		4 Numerical Results
			4.1 Experimental Convergence Rates
			4.2 Numerical Validation of the Entropy and Kinetic Energy Analysis
		5 Conclusions
		References
	Explicit Modal Discontinuous Galerkin Approximations for Three-Dimensional Electronic Boltzmann Transport Equation
		1 Introduction
		2 Governing Equations
		3 Numerical Procedure
		4 Results and Discussion
			4.1 Accuracy Test
			4.2 Dynamics of Three-Dimensional Electronic BTE
		5 Conclusion
		References
	High Order Compact Central Spatial Discretization Under the Framework of Entropy Split Methods
		1 Introduction and Background
			1.1 SBP Boundary Closure
			1.2 Application to the Equations of Gas Dynamics and MHD
		2 Numerical Examples
		3 Summary
		Appendix
		References
	High Order Solution to Exterior 3D Wave Equation by the Method of Difference Potentials
		1 Introduction
		2 Fourth Order Compact Scheme (FOCS)
		3 Method of Difference Potentials (MDP)
			3.1 Preliminaries
			3.2 Boundary Equation with Projection (BEP)
			3.3 Solving the Boundary Equation with Projection
		4 Numerical Simulations
		5 Concluding Remarks and Future Work
		Appendix 1: Extension Operator
		Appendix 2: Generating Test Solutions
		References
	Dual-Primal Isogeometric Tearing and Interconnecting Methods for the Stokes Problem
		1 Introduction
		2 Problem Formulation and IETI-DP
		3 Numerical Experiments
		4 Conclusions and Final Remarks
		References
	Mimetic Relaxation Runge Kutta Methods
		1 Introduction
		2 Overview of Mimetic and Relaxation RK Methods
		3 Numerical Results
			3.1 Example 1
			3.2 Example 2: System of Hyperbolic Equations
			3.3 Example 3: Non-linear Burgers\' Equation
		4 Conclusion
		References
	DoD Stabilization for Higher-Order Advection in Two Dimensions
		1 Introduction
		2 Problem Setup
		3 Stabilization Terms
			3.1 L2 Stability for Semi-discrete Scheme
		4 Numerical Results
		5 Conclusion
		References
	Space-Time Error Control Using a Partition-of-Unity Dual-Weighted Residual Method Applied to Low Mach Number Combustion
		1 Introduction
		2 The Low Mach Number Combustion Equations
		3 Discretization
		4 Space-Time PU-DWR Error Estimation
			4.1 Error Estimation
			4.2 Practical Evaluation
			4.3 Variational PU Localization
		5 Numerical Example
			5.1 Comparison of Error Estimators
			5.2 Adaptive Results
		6 Conclusions
		References
	Eigenspectral Analysis of Preconditioners in an Adaptive Compressible Flow Solver
		1 Introduction
		2 Adaptive Compressible Flow Solver
			2.1 Spatial Discretization
			2.2 Implicit Time Integration
			2.3 Balanced Adaptive Time Stepping
		3 Eigenspectral Analysis of Different Preconditioners
			3.1 Different Preconditioners
			3.2 Eigenspectral Analysis
			3.3 Numerical Tests
		4 Conclusion
		References
	Comparative Study on a Variety of Structure-Preserving High Order Spatial Discretizations with the Entropy Split Methodsfor MHD
		1 Introduction
		2 The Symmetrizable Ideal MHD Equations Godunov-MHD-source-term,36:Sjogreen-Yee-18,Yee-sjogreen-MHD-2005
			2.1 ECLOG for MHD
			2.2 ECLOGKP for MHD
			2.3 ECHKP for MHD
			2.4 Ducros Split (DS) for MHD
			2.5 Ducros Split with KEP for MHD (DSKP)
			2.6 KGP Split for MHD
			2.7 Entropy Split Scheme (ES) for MHD
		3 Numerical Experiments
			3.1 2D Alfvén Wave
			3.2 1D MHD Brio-Wu Riemann Problem Brio-Wu
		References
	High-Order Discretisations and Efficient Direct Space-Time Finite Element Solvers for Parabolic Initial-Boundary Value Problems
		1 Introduction
		2 Kronecker Product
		3 Space-Time Finite Element Spaces
		4 Space-Time Finite Element Method
		5 Space-Time Solver
		6 Numerical Examples
		7 Conclusions
		References
	A Subcell Limiting Based on High-Order Compact Nonuniform Nonlinear Weighted Schemes for High-Order CPR Method
		1 Introduction
		2 Review of High-Order CPR and High-Order CNNW
			2.1 High-Order CPR
			2.2 High-Order CNNW
		3 A Priori Subcell CNNW Limiting Approach for CPR Method
			3.1 Troubled Cell Indicator
			3.2 Subcell Limiting Based on CNNW
		4 Numerical Investigation
			4.1 Shu-Osher Problem
			4.2 2D Riemann Problem
			4.3 Shock-Vortex Interaction
		5 Concluding Remarks
		References